Linear Algebra - Two questions

1. Sep 14, 2011

MJay82

1. The problem statement, all variables and given/known data
Q1: Find the value(s) of h for which the vectors are linearly dependent. Justify your answer.
Q2: The linear transformation T is defined by T(x)=Ax. Find a vector x whose image under T is b and determine whether x is unique.

2. Relevant equations
Q1: I'm going to write the vectors as linear equations instead, since it should be easier to input:
1(x1) -2(x2) + 3(x3)
5(x1) -9(x2) + h(x3)
-3(x1) +6(x2) -9(x3)

Q2: I will write Matrix A with vector b in linear equation form as well.
1(x1) -5(x2) -7(x3) = -2
-3(x1) +7(x2) +5(x3) = -2

3. The attempt at a solution
I feel like I'm mostly right on these, but I would like some confirmation before I have to turn them in.

For Q1: I noticed that row 3 is a scalar multiple of row 1, so I performed the necessary row operation to make it a zero row. Then I replaced row 2 with the sum of row 2 and (-5) row 1. This left me with:
1(x1) - 2(x2) + 3(x3)
0(x1) + 1(x2) +h-15(x3)
0 0 0

I'm a little fuzzy on linear dependency, but I thought that if I could make x3 a free variable, then I'd have it. But then I noticed that x3 is always going to be a free variable, so it seems to me that the solution should be all real numbers.

For Q2 - I'll just say the row operations I did, and the solution that I came up with:
(3)row 1 + row 2 replace row 2.
Scale row 2 by -(1/8)
(5)row 2 + row 1 replace row 1.

This left me with:
(x3) free
(x1) = 3 - 3(x3)
(x2) = 1 - 2(x3)
And my answer is: Since (x3) is free, the solution is not unique.
Thanks for any help.

Last edited: Sep 14, 2011
2. Sep 14, 2011

flyingpig

Take the determinant of the matrix and set it to 0

3. Sep 14, 2011

Staff: Mentor

These are NOT equations. If they were, each row would have an = in it.

What are x1, x2, and x3? Are they vectors? Are they components of a single vector?

Since you don't have a system of equations, what you're doing here doesn't seem valid to me.

What is the exact wording of these problems?

4. Sep 15, 2011

MJay82

The exact wording of the problems is what I wrote.
For Q1, just imagine a coeffecient matrix, now imagine them listed as column vectors V1-V3 instead.