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Multiplying a Vector by a Matrix Worksheets

In a way a vector is a matrix because it can have a single row and column. As a result, we can perform a variety of operations between these two methods for modelling data. These worksheets will help you understand how to find the products of a vector and a matrix. The math that is involved here is straight forward. Where students get lost in the organization of it all. They don’t know what they are multiplying by what. This why is it paramount to write your problems clearly and stay organized through the entire process of your calculation. The lessons you find below will show you how step by step.

Aligned Standard: HSN-VM.C.11

  • Answer Keys - These are for all the unlocked materials above.

Homework Sheets

This is where I usually have students draw lines to show how to process the product of the values.

  • Homework 1 - The first matrix (c) has 3 rows. The second matrix (x) has 1 column. So the product will have 3 rows and 1 column (3 x 1).
  • Homework 2 - You need to figure out the number of rows and columns the product will have. This is determined by the number of rows of the first matrix.
  • Homework 3 - We have many three by ones here.

Practice Worksheets

These are orientated slight differently than the homework. I have seen this format on several different assessments.

  • Practice 1 - Find the matrix product of C and vector x.
  • Practice 2 - Find the product of the vector and the matrix.
  • Practice 3 - This is a technique and calculation that computers process quicker than you could read them.

Math Skill Quizzes

Test out your skills on this topic.

  • Quiz 1 - The key is understand every element in both the vector and matrix.
  • Quiz 2 - This introduces you to the vector outer product technique.
  • Quiz 3 - I would spend a good amount of time pushing in on these.

How to Multiply a Vector by a Matrix?

Multiplication between a matrix and a vector is very common. In order to learn about how multiplication between a matrix A and a vector X, also known as matrix-vector product, all we need to do is view vector X as a column matrix. In order for this to work they both need the same dimensions (number of rows and columns).

There are actually two different instances of multiplication taken place when we do this. There is an inner and outer form of multiplication. The inner form results in each term of the vector being multiplied by each term of the vector and each set of terms in one direction are summed to create a new term. This is the operation as the dot product between vectors and standard multiplication applies. The outer form of multiplication each term in the vector is multiplied by each term in the matrix.

Therefore, if A is a vector m x n matrix (i.e., multiplied with n columns), the product Ax is defined by n x 1. If we let A x = b, then b is m x 1 column-vector. To put it simply, the number of rows in A determines the number of rows in product b. It may confuse you in the first look, but the process is quite simple. It only takes the dot product of x to be multiplied with each of the row of A. This is the reason the number of columns in A has to be equal the number of elements/components in vector x.

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