Range of Left Multiplication Matrix

[nautolian]

Homework Statement

Consider the left multiplication L_A:R^4->R^3 corresponding the matrix A:

1 2 -1 3 = 2
2 4 -1 6 = 5
0 1 0 2 = 3

What is the rank of L_A and the Range of L_a?

Homework Equations

The Attempt at a Solution

I have two main problems with this question. First, what is the left multiplication? I can't seem to find it anywhere and the prof said that we should know it from previous classes, but I have never heard of the left multiplication. Secondly, I understand that that the Range is supposed to be the solution space for Ax=B: is that correct?
Thank you so much for your help in advance.

 

[Mark44]

Homework Statement

Consider the left multiplication L_A:R^4->R^3 corresponding the matrix A:

1 2 -1 3 = 2
2 4 -1 6 = 5
0 1 0 2 = 3

Why are the = signs in there?

This is A.
1 2 -1 3
2 4 -1 6
0 1 0 2

What is the rank of L_A and the Range of L_a?

Homework Equations

The Attempt at a Solution

I have two main problems with this question. First, what is the left multiplication?

Edit: L_A is a transformation whose matrix representation is A. L_A takes a vector in R⁴ as input, and produces a vector in R³.

The transformation L_A corresponds to A times a vector. Here A is on the left.

I can't seem to find it anywhere and the prof said that we should know it from previous classes, but I have never heard of the left multiplication. Secondly, I understand that that the Range is supposed to be the solution space for Ax=B: is that correct?

No. In the equation Ax = b, the range of A is the set {b | b = Ax}

Capital letters are usually used to represent matrices, and lower case letters are usually used to represent vectors.

Thank you so much for your help in advance.

 

[nautolian]

Oh, I see. Thank you. So for the Range, because they give you what the system of equations equals, would that be your b? or would you solve for Ax=b where b is, say, (x,y,z)?
Thanks again.

 

[Mark44]

Your problem statement isn't very clear. Are they asking whether <2, 5, 3>^T is in the range of L_A?

Or are they asking what is the range of L_A?

or would you solve for Ax=b where b is, say, (x,y,z)?

Your notation is confusing, because in your first equation x is a vector, while in b, x is the first component.

To find the range, write the matrix equation Ax = b as an augmented matrix like so: [A | b], where b = <b₁, b₂, b₃>^T. Then row-reduce this augmented matrix. What you're finding is that for any vector b, is there a vector x such that Ax = b?