Linear Algebra - Two questions

[MJay82]

Homework Statement

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.

Homework 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

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.

 

[flyingpig]

Take the determinant of the matrix and set it to 0

 

[Mark44]

Homework Statement

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.

Homework 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)

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?

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

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

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?

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.

 

[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.