HSN.VM.C.11Common CoreMathNumber and QuantityGrades 9-12
HSN.VM.C.11: Multiplying Vectors by Matrices and Matrices as Transformations
In plain English: HSN.VM.C.11 is an advanced (+) Common Core number and quantity standard that asks students to multiply a vector, written as a one-column matrix, by a matrix of suitable dimensions to get a new vector, and to treat a matrix as a rule that transforms vectors, for example by rotating, reflecting or stretching them. It is usually taught in Precalculus.
(+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
Common Core State Standards for Mathematics · Domain: Vector and Matrix Quantities (VM) · Cluster: Perform operations on matrices and use matrices in applications. Also written as HSN-VM.C.11 or N-VM.11 · Official standard
Students learn to multiply a vector by a matrix. A vector with n components is treated as an n × 1 matrix, a matrix with one column. An m × n matrix can multiply it only when the matrix has as many columns as the vector has entries, and the result is a new vector with m entries. Each entry of the product is one row of the matrix times the vector: multiply matching entries and add. Students also see the same product column by column: Av is the first column of A times the first entry of v, plus the second column times the second entry, and so on.
The second half of the lesson treats a matrix as a transformation: an input vector goes in and an output vector comes out. Students apply matrices that rotate, reflect, stretch and shear vectors in the plane, find the matrix for a described transformation from where it sends [1; 0] and [0; 1], apply two transformations in a row, and use a matrix to move a population vector forward one year. Matrices and vectors are written row by row with semicolons between rows: [3 1; -2 4] has first row 3, 1 and second row -2, 4, and the column vector [2; 5] has 2 on top of 5.
Learning Objectives
By the end of this lesson, students will be able to:
Decide from the dimensions whether a matrix-vector product is defined and state the size of the result
Multiply a vector by a 2 × 2, 3 × 3 or non-square matrix, row by row, and interpret the result as a combination of the columns
Apply a matrix to vectors in the plane and describe the transformation (rotation, reflection, dilation, stretch or shear)
Find the matrix of a transformation from the images of [1; 0] and [0; 1], and apply two transformations in a row
Use a matrix to transform a data vector, such as a population split between two regions, and interpret the new vector
Prior Knowledge Required
Students should already be comfortable with:
Vectors in component form and drawing them as arrows HSN.VM.A.2
Multiplying a vector by a scalar HSN.VM.B.5
Transformations as functions that take points to points HSG.CO.A.2
Matrix dimensions (rows × columns) and naming entries by row and column
Write a rule on the board and ask students to apply it before any matrices appear.
Warm-Up Prompt
"A rule takes the point (x, y) to the point (2x + y, x - 3y). Where does (4, 1) go? Where does (1, -2) go? Which point goes to (0, 0)?"
Collect answers: (4, 1) goes to (9, 1), and (1, -2) goes to (0, 7). Only (0, 0) goes to (0, 0). Then write the coefficients of the two expressions as the rows of a matrix, [2 1; 1 -3], and show that multiplying it by the column vector [4; 1] row by row gives [9; 1], the same answer. Tell students that every matrix is a rule of this kind, and that the lesson is about computing with it and seeing what it does to vectors.
Direct Instruction20-25 minutes
Part 1: The product. Show the dimension rule first: an m × n matrix times an n × 1 vector gives an m × 1 vector. The inside numbers must match, and the outside numbers give the size of the answer. Then show the two ways of computing, row by row and column by column (Diagram 2).
Check dimensions: count the columns of the matrix and the entries of the vector. They must be equal.
Row by row: entry i of the product is row i of the matrix times the vector, entry by entry, added up.
Column check: the product also equals (first entry of v) × (column 1) + (second entry of v) × (column 2) + ..., which is a quick way to check the answer.
Read the result: the output is a vector with as many entries as the matrix has rows.
Multiply the 2 × 3 matrix B = [1 0 2; -1 3 1] by w = [4; -2; 3]. Can B multiply [4; -2]?
Equation: Bw = [4 + 0 + 6; -4 - 6 + 3] = [10; -7], a 2 × 1 vector; B[4; -2] is not defined (3 columns, 2 entries)
Rotation of a vector
Apply R = [0 -1; 1 0] to v = [3; 1] and compare the lengths (Diagram 1).
Equation: Rv = [-1; 3]; both vectors have length √10, and Rv is v turned 90° counterclockwise
Matrix from the images of [1; 0] and [0; 1]
Find the matrix that reflects every vector across the line y = x, then apply it to [5; -2].
Equation: [1; 0] goes to [0; 1] and [0; 1] goes to [1; 0], so the matrix is [0 1; 1 0], and [0 1; 1 0][5; -2] = [-2; 5]
Transforming a data vector
Each year 80% of city residents stay and 20% move to the suburbs, while 10% of suburban residents move to the city and 90% stay. Start with 500 thousand in the city and 300 thousand in the suburbs.
Equation: [0.8 0.1; 0.2 0.9][500; 300] = [430; 370]: 430 thousand in the city and 370 thousand in the suburbs after one year
Part 2: Matrices as transformations. After the third example, use Diagram 1 to compare two matrices acting on the same vector. Then explain why the fourth example works: A[1; 0] is the first column of A and A[0; 1] is the second column, so a matrix is completely determined by where it sends these two vectors. For the fifth example, note that each column of the matrix adds to 1, so the total population, 800 thousand, stays the same; the matrix only moves people between regions. Finish Part 2 by applying two transformations in a row: reflect [3; 1] across the x-axis to get [3; -1], then rotate that by R to get [1; 3].
Guided Practice15 minutes
Pairs work through four problems, one at a time, and compare answers with another pair before moving on.
(c) For the 2 × 3 matrix [1 2 3; 4 5 6], which of [1; 1] and [1; 0; -1] can it multiply? (Only the second, giving [-2; -2].)
(d) Apply [0 1; -1 0] to [2; 5] and to [1; 0]. Draw the arrows and describe the transformation. (Answers: [5; -2] and [0; -1]: a rotation of 90° clockwise.)
Listen for these errors: multiplying down columns of the matrix instead of across rows, writing the answer as a matrix of the same size as A, and trying to multiply when the dimensions do not match.
Independent Practice15 minutes
Students complete five problems on their own and check each product with the column picture.
[-1 4; 2 0][3; -2] (answer: [-11; 6])
[1 1 1][4; -2; 7], a 1 × 3 matrix times a 3 × 1 vector (answer: the 1 × 1 matrix [9], the sum of the entries)
Write the matrix that triples every vector and apply it to [-2; 5] (answer: [3 0; 0 3], giving [-6; 15])
Reflect [6; 4] across the y-axis with [-1 0; 0 1] (answer: [-6; 4])
Apply S = [1 2; 0 1] twice to [1; 1] (answers: S[1; 1] = [3; 1], then S[3; 1] = [5; 1])
Closure5 minutes
Exit ticket: (1) A 3 × 2 matrix multiplies a vector. How many entries must the vector have, and how many will the result have? (2 and 3.) (2) Compute [2 0; -1 3][1; 4]. (Answer: [2; 11].) (3) Describe what [-1 0; 0 -1] does to every vector in the plane. (It reverses the direction and keeps the length, a rotation of 180°.)
Differentiation Strategies
For Struggling Students
Cover all rows but one with a strip of paper, so that each entry of the product is computed from a single row
Write the dimensions under every matrix and vector (for example 2 × 3 and 3 × 1) and circle the inside numbers before multiplying
Draw every input and output vector on grid paper, so that a transformation's effect can be seen and not only computed
For Advanced Students
Ask students to find the matrix that rotates vectors by 45° counterclockwise and to check that it keeps the length of [1; 0]
Ask for a matrix that sends every vector in the plane onto the x-axis, and discuss which input vectors go to the zero vector
Ask students to show that applying [1 0; 0 -1] and then [0 -1; 1 0] gives the same result on every vector as a single matrix, and to find that matrix (this previews matrix multiplication in HSN.VM.C.8)
Assessment Guidance
What to Look For
Check that students state dimensions before multiplying and can explain why a product is or is not defined. Look for row-by-row products written with the right signs, and ask students to confirm one answer with the column picture. For transformations, ask students to draw the input and output arrows and name the effect, and to find a matrix by asking where it sends [1; 0] and [0; 1]. For data vectors, check that students say what each entry of the output means in context.
02
Classroom Activities
3 Activities
1
Transformation Stations
20 minGroups of 3-4
Four stations each show one 2 × 2 matrix. Groups rotate through the stations, apply the station's matrix to three test vectors, draw the results and name the transformation.
Stations
Station 1: [1 0; 0 -1], reflection across the x-axis
Station 2: [0 -1; 1 0], rotation of 90° counterclockwise
Station 3: [2 0; 0 2], dilation by a factor of 2
Station 4: [1 0; 1 1], a vertical shear
Procedure
At every station, multiply the matrix by [1; 0], [0; 1] and [2; 3], and draw each input and output arrow on grid paper in two colors
Expected outputs for [2; 3]: Station 1 gives [2; -3], Station 2 gives [-3; 2], Station 3 gives [4; 6] and Station 4 gives [2; 5]
Write one sentence describing the effect, then check your sentence against the images of [1; 0] and [0; 1]: they are the columns of the matrix
Discussion Questions
Which stations keep the length of every vector? How can you tell from the outputs?
At Station 4, which vectors do not change? Why?
Which station's matrix would undo Station 2?
Modification for Distance Learning
Use a free graphing tool with a slider-free vector plot: students type each input and output vector and share a screenshot of each station's arrows.
2
Migration Model
20 minPairs
Pairs use a matrix to move a population vector forward three years and look for a pattern. The matrix transforms this year's vector into next year's.
Setup
A region has a city and its suburbs. Each year 85% of city residents stay and 15% move to the suburbs; 5% of suburban residents move to the city and 95% stay
Transition matrix: P = [0.85 0.05; 0.15 0.95]. Starting vector, in thousands: [600; 400] (city; suburbs)
Procedure
Partner A computes the next year's vector by hand; Partner B checks with a calculator; swap roles each year
Expected vectors: year 1 [530; 470], year 2 [474; 526], year 3 [429.2; 570.8]
Add the two entries each year and explain why the total stays 1,000 thousand
Discussion Questions
What does the entry 0.15 mean, and why is it in row 2, column 1?
Will the city population keep falling forever? Predict, then test by computing five more years with the calculator.
3
Dimension Dominoes
15 minGroups of 3
Groups receive a set of cards with matrices and one starting vector. They chain the cards so that every product is defined, compute each output vector and record how its size changes.
Cards
Start: the vector [2; -1] (2 × 1)
Card K: [1 2; 0 -1; 3 1] (3 × 2)
Card L: [1 0 -2; 2 1 1] (2 × 3)
Card N: [1 -1] (1 × 2)
Procedure
Find the only order in which all three cards can be used: K, then L, then N
Compute each step: K[2; -1] = [0; 1; 5], then L[0; 1; 5] = [-10; 6], then N[-10; 6] = [-16]
Record the sizes: 2 × 1, then 3 × 1, then 2 × 1, then 1 × 1
Challenge Variation
Groups design a new card that could be used after Card N and one that could be used before Card K, then trade card sets with another group.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Two Matrices Acting on the Same Vector
The vector v = [3; 1] (navy) and its images under the rotation R = [0 -1; 1 0] (blue) and the reflection F = [1 0; 0 -1] (dashed gray), drawn to scale on a grid with 1 unit per square. Both matrices keep the length √10, but they send v to different vectors.
Diagram 2: A Matrix Times a Vector, Row by Row and Column by Column
For A = [2 -1; 1 1] and v = [1; 2], one copy of column 1 plus two copies of column 2 lands at [0; 3], the same vector the row-by-row computation gives. The grid is drawn to scale.
04
Homework Assignment
~30 min
HSN.VM.C.11 Homework: Matrix-Vector Products and Transformations
Directions: Write the dimensions under every matrix and vector before multiplying. Show each row-by-row product. For Part 2, draw every input and output vector on grid paper. Write matrices row by row with semicolons between rows. A calculator may be used for Problem 5.
Part 1: Multiplying Vectors by Matrices (Problems 1-2)
A is 3 × 2 and B is 2 × 4. (a) For each of the vectors [1; 2], [1; 2; 3] and [1; 2; 3; 4], say whether A can multiply it, whether B can multiply it, and the size of each defined result. (b) A 4 × 3 matrix times a vector u gives a vector w. What are the sizes of u and w?
Part 2: Matrices as Transformations (Problems 3-4)
Let K = [3 0; 0 1]. (a) Apply K to [1; 2], [-2; 1] and [0; -3]. (b) Draw each input and output and describe what K does. (c) Which vectors does K leave unchanged?
(a) Find the matrix that reflects every vector across the line y = -x, by deciding where it sends [1; 0] and [0; 1]. (b) Apply it to [4; 1]. (c) Apply it again to your answer. What do you notice, and why does it make sense?
Part 3: Applications (Problems 5-6)
Each week, 70% of students who buy school lunch buy it again the next week and 30% switch to bringing lunch; 20% of students who bring lunch switch to buying and 80% keep bringing it. This week 300 students buy and 200 bring. (a) Write the transition matrix with columns for this week's choice. (b) Find next week's vector and the vector for the week after. (c) Explain what each entry of your answers means.
In a video game, a character moves with velocity vector [5; -2] (pixels per frame). A left turn multiplies the velocity by [0 -1; 1 0], and a speed boost multiplies it by [1.5 0; 0 1.5]. (a) Find the velocity after one left turn and after two left turns. (b) Show that a left turn does not change the speed. (c) Find the velocity after one left turn followed by one boost.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Dimensions
Every product checked for size, defined and undefined cases explained
Sizes stated but not explained
Dimensions ignored
Products
All row-by-row products correct, one checked by columns
One or two arithmetic errors
Most products incorrect
Transformations
Inputs and outputs drawn, effect named, matrix found from images of [1; 0] and [0; 1]
Effect named without drawings or reasoning
Missing or incorrect
Interpretation
Output vectors explained in context with units
Some interpretation
No interpretation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. Matrices and vectors are written row by row, as in the lesson plan.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Compute [2 -1; 4 3][3; 2].
Answer: B
Row by row: 2(3) + (-1)(2) = 4 and 4(3) + 3(2) = 18, so the product is [4; 18]. Choice A multiplies the vector by the columns instead of the rows. Choice C multiplies each column by one entry and never adds. Choice D loses the minus sign in row 1.
Question 2 of 20 · Multiple Choice
A 3 × 4 matrix multiplies a vector. What size must the vector be, and what size is the product?
Answer: D
The number of columns of the matrix (4) must equal the number of entries of the vector, and the product has one entry per row of the matrix (3). Choice A reverses both numbers. Choice B uses the column count for the size of the answer.
Question 3 of 20 · Multiple Choice
Which product is not defined?
Answer: A
In choice A, the matrix has 3 columns but the vector has only 2 entries, so the rows cannot be matched with the vector. Choice C is a 3 × 2 matrix times a 2 × 1 vector, which is defined and gives a 3 × 1 vector. Choice D gives the 1 × 1 matrix [32].
Question 4 of 20 · Multiple Choice
Compute [1 0 2; 0 3 -1][3; 2; -1].
Answer: A
Row 1: 1(3) + 0(2) + 2(-1) = 1. Row 2: 0(3) + 3(2) + (-1)(-1) = 7. The product is the 2 × 1 vector [1; 7]. Choice D forgets that (-1)(-1) is +1. Choice C has three entries, but a 2 × 3 matrix always produces a vector with 2 entries.
Question 5 of 20 · Multiple Choice
What does the matrix [1 0; 0 0] do to every vector in the plane?
Answer: B
[1 0; 0 0][x; y] = [1x + 0y; 0x + 0y] = [x; 0], so every vector is flattened onto the x-axis (a projection). Choice A would be [x; -y]. Choice C mixes up which row gives which component, and choice D would need the identity matrix [1 0; 0 1].
Question 6 of 20 · Multiple Choice
Apply R = [0 -1; 1 0] to the vector [5; -1].
Answer: D
Row 1: 0(5) + (-1)(-1) = 1. Row 2: 1(5) + 0(-1) = 5. So R[5; -1] = [1; 5], the vector turned 90° counterclockwise. Choice A drops the sign in row 1: (-1)(-1) is +1 (it is also what you get by only swapping the entries, a reflection across y = x). Choice C only changes the sign of the second entry, which is a reflection across the x-axis, not a rotation.
Question 7 of 20 · Multiple Choice
A matrix sends [1; 0] to [2; 1] and [0; 1] to [-1; 3]. Which matrix is it?
Answer: C
A matrix times [1; 0] is its first column and times [0; 1] is its second column, so the images are the columns: [2 -1; 1 3]. Choice A writes the images as rows. Check: [2 1; -1 3][1; 0] = [2; -1], not [2; 1].
Question 8 of 20 · Multiple Choice
What does the matrix [5 0; 0 5] do to every vector?
Answer: A
[5 0; 0 5][x; y] = [5x; 5y] = 5[x; y], a dilation by 5. Choice B confuses multiplying with adding. Choice C multiplies the two diagonal entries, but only one factor of 5 applies to each component.
Question 9 of 20 · Multiple Choice
Which expression equals [3 1; 2 -1][2; 1]?
Answer: C
In the column picture, the first entry of the vector multiplies column 1, [3; 2], and the second entry multiplies column 2, [1; -1]: 2[3; 2] + 1[1; -1] = [7; 3]. Choice A uses the rows as if they were columns, and choice D multiplies only the diagonal entries.
Question 10 of 20 · Multiple Choice
Each year, 90% of a town's residents stay and 10% move to a nearby city, while 20% of city residents move to the town and 80% stay. The transition matrix is P = [0.9 0.2; 0.1 0.8]. This year the town has 600 residents and the city 900. What is next year's vector (town; city)?
Answer: B
P[600; 900] = [0.9(600) + 0.2(900); 0.1(600) + 0.8(900)] = [540 + 180; 60 + 720] = [720; 780]. The total, 1,500, is unchanged. Choice A keeps only the people who stayed and forgets the ones who moved in. Choice D swaps the two entries.
Question 11 of 20 · Multiple Choice
Which matrix doubles the length of every vector and reverses its direction?
Answer: D
[-2 0; 0 -2][x; y] = [-2x; -2y] = -2[x; y], which points the opposite way and is twice as long. Choice A keeps the direction. Choice B changes the sign of only the first component, which is a reflection followed by a stretch.
Question 12 of 20 · Multiple Choice
S = [1 1; 0 1]. Apply S to [2; 3], then apply S again to the result.
Answer: C
S[2; 3] = [2 + 3; 3] = [5; 3], and S[5; 3] = [5 + 3; 3] = [8; 3]. Choice A stops after one step. The shear adds the second component to the first each time and never changes the second component, so choices B and D cannot be right.
Question 13 of 20 · Multiple Choice
Which vector is left unchanged by the matrix [0 1; 1 0]?
Answer: D
[0 1; 1 0][x; y] = [y; x] swaps the components, a reflection across the line y = x. Only vectors with equal components stay the same, such as [4; 4]. Choice A goes to [-4; 4], and choices B and C swap into each other.
Question 14 of 20 · Multiple Choice
A matrix M takes every vector with 5 entries to a vector with 2 entries. What are the dimensions of M?
Answer: B
M times a 5 × 1 vector must be 2 × 1, so M needs 5 columns (to match the input) and 2 rows (one per output entry): 2 × 5. Choice A reverses rows and columns; a 5 × 2 matrix would take 2 entries to 5.
Write the matrix that stretches every vector horizontally by a factor of 4 and also reflects it across the x-axis. Explain how you found it, and apply it to [-3; 7].
The transformation sends [1; 0] to [4; 0] and [0; 1] to [0; -1], and those images are the columns: [4 0; 0 -1]. Then [4 0; 0 -1][-3; 7] = [-12; -7].
Question 17 of 20 · Short Answer
Commuters in a city ride a bike or take the bus. Each month 60% of bike riders keep biking and 40% switch to the bus; 30% of bus riders switch to biking and 70% keep riding the bus. This month 200 people bike and 100 take the bus. Use the matrix [0.6 0.3; 0.4 0.7] to find the vectors (bike; bus) for the next two months.
Next month: [0.6(200) + 0.3(100); 0.4(200) + 0.7(100)] = [150; 150]. The month after: [0.6(150) + 0.3(150); 0.4(150) + 0.7(150)] = [135; 165]. The total stays 300.
Question 18 of 20 · Short Answer
Let A = [3 -4; 4 3]. (a) Find A[1; 0], A[0; 1] and A[2; 1]. (b) Compare the length of [2; 1] with the length of A[2; 1]. What does A seem to do to vectors?
(a) [3; 4], [-4; 3] and [2; 11]. (b) |[2; 1]| = √5 and |[2; 11]| = √125 = 5√5, so the length is multiplied by 5. The images of [1; 0] and [0; 1] both have length 5 and are perpendicular, so A rotates vectors (by about 53°) and stretches them by a factor of 5.
Question 19 of 20 · Short Answer
Explain why [2 0 1; -1 3 2][1; 1] is not defined. Then compute [2 0 1; -1 3 2][0; 2; 1].
The matrix is 2 × 3, so it needs a vector with 3 entries; [1; 1] has only 2. With [0; 2; 1]: row 1 gives 0 + 0 + 1 = 1 and row 2 gives 0 + 6 + 2 = 8, so the product is [1; 8].
Question 20 of 20 · Short Answer
A matrix M sends [1; 0] to [0; 2] and [0; 1] to [1; 0]. Find M and compute M[3; -4].
The images are the columns, so M = [0 1; 2 0]. Then M[3; -4] = [0(3) + 1(-4); 2(3) + 0(-4)] = [-4; 6].
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSN.VM.C.11 mean?
HSN.VM.C.11 asks students to multiply a vector by a matrix and to understand the result as a new vector. The vector is treated as a matrix with one column, the matrix must have as many columns as the vector has entries, and the product has one entry for each row of the matrix. The second sentence of the standard asks students to see a matrix as a transformation: a rule that takes each input vector to an output vector, for example by rotating, reflecting or stretching it.
Is HSN.VM.C.11 taught in Algebra 2 or Precalculus?
It is usually taught in Precalculus. The (+) marks it as additional mathematics for students who take advanced courses. It often comes right after matrix addition and multiplication (HSN.VM.C.8) and before 2 × 2 matrices as transformations of the whole plane (HSN.VM.C.12).
How do you multiply a matrix by a vector?
Go row by row. For each row of the matrix, multiply its entries by the matching entries of the vector and add. That sum is one entry of the answer. For example, [3 1; -2 4][2; 5] has first entry 3(2) + 1(5) = 11 and second entry -2(2) + 4(5) = 16, so the product is [11; 16].
When can a matrix multiply a vector?
Only when the number of columns of the matrix equals the number of entries of the vector. An m × n matrix times an n × 1 vector gives an m × 1 vector. So a 2 × 3 matrix can multiply a vector with 3 entries, and the answer has 2 entries; it cannot multiply a vector with 2 entries.
What does "a vector regarded as a matrix with one column" mean?
The standard asks students to write a vector such as ⟨4, -2, 3⟩ as a column, [4; -2; 3], which is a 3 × 1 matrix. Written this way, the vector follows the same multiplication rule as any matrix, and the product of a matrix and a vector is again a column, so it is again a vector.
What does it mean that a matrix transforms vectors?
A matrix A acts like a function: input a vector v and the output is Av. In the plane, you can draw v and Av as arrows and describe the effect. [0 -1; 1 0] turns every vector 90° counterclockwise, [1 0; 0 -1] reflects it across the x-axis, and [2 0; 0 2] doubles its length. With data, a matrix can transform this year's population vector into next year's.
How do you find the matrix for a given transformation?
Decide where the transformation sends [1; 0] and [0; 1]. Those two output vectors are the columns of the matrix, in that order. For a reflection across the line y = x, [1; 0] goes to [0; 1] and [0; 1] goes to [1; 0], so the matrix is [0 1; 1 0].
What are common mistakes with matrix-vector products?
A common error is multiplying down the columns of the matrix instead of across its rows. Others are trying to multiply when the dimensions do not match, losing signs such as (-1)(-3) = 3, writing the image vectors of [1; 0] and [0; 1] as rows instead of columns, and writing the answer as a matrix instead of a vector. The column picture is a fast check on any product.
Where are matrix transformations of vectors used?
Computer graphics and video games use matrices to rotate, scale and reflect the vectors that describe positions and velocities. Population and market-share models use a transition matrix to move a vector of counts forward one step at a time. Physics and engineering use matrices to change coordinates. In each case the same row-by-row product does the work.
How does HSN.VM.C.11 connect to other standards?
It builds on vectors and scalar multiplication (HSN.VM.A.2 and HSN.VM.B.5) and on transformations as functions (HSG.CO.A.2). The product Av is exactly what students write when they turn a system into a matrix equation AX = B (HSA.REI.C.8). HSN.VM.C.12 extends the idea from single vectors to the whole plane and connects the determinant to area.
07
Related Standards
5 standards
These standards connect to HSN.VM.C.11: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSN.VM.B.5Prerequisite
Multiply a vector by a scalar, graphically and component-wise
