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In Exercises $25-28$ , determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. The transformation in Exercise 2

a. $A=\left[\begin{array}{ccc}{1} & {4} & {-5} \\ {3} & {-7} & {4}\end{array}\right]$b. Yes

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 9

The Matrix of a Linear Transformation

Introduction to Matrices

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So it's the matrix that we got from exercise, too. And now we want to see if the mapping ISS want one are onto or both. Also for party wanted one. We know that the, uh, mapping is wonder one if all the columns of a r linearly independent. But since we have a two by three matrix, we have more columns than we do, rose. So the columns definitely have to be linearly independent. Um, so we do have a wonder one napping to check. If it is on to, we need to see if the columns of a span, um are too. So ah, three and four. I believe in the book tells us that the columns of a span are too. If a has a pivot position in every bro. So we're going to start reducing this matrix, so ah, we want there to be a zero here, so I'm gonna take three times the first row and subtract the second row. So our first row stays the same. So one times three is three minus 30 Four times three is 12 minus a negative. Seven is 19 negative. 19 uh, negative. Three times five is 15 15 plus four gives us 19. So now we're in echelon form so we can keep going, but the pivots aren't gonna change. So we have a pivot here, and we're gonna have a pivot here. So since there's one in each row, the columns span are too. So this tells us that the mapping is on two.

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