Linear Algebra: Parametric Solution Set

[Ty Ellison]

Homework Statement

[/B]
Suppose the solution set of some system Ax = b , Where A is a 4x3 matrix, is

*Bold characters are vectors*

x_1= 1 + 3t
x_2 = 2 - t
x_3 = 3 + 2t

where t is a parameter and can be any number.

a) How many pivots are in the row echelon form of A?

b) Let u, v, w be the columns of A. Do they span the whole R^4? Explain.

c) Is u, v, w a linearly independent set? Explain.

d) Produce a linear relation for u, v, w. ( Hint: what is the solution of the system Ax = 0 ?)​

Homework Equations

The Attempt at a Solution

My original attempt I tried to solve for the parameter t in hopes that this would give me a coefficient matrix for x that I could substitute in the equation for x but I could not make this work. I also realized that the solution set is linearly dependent as one of the rows can be expressed as linear combinations of the other rows but I don't know if this is significant information for this problem.

My secondary attempt was to consider a system where the solution set in the form listed above was b, then A would be a 4x3 identity matrix. I then multiplied each side of the equation by the 4x3 Identity matrix but multiplying the equation by 1 was not leading me to the next path, and thus now I sit here in utter confusion trying to make sense of this problem and is making me question my confidence in my other solutions on the assignment .

Note: I am having trouble mainly with formulating an expression or method to solve Part A, but I feel with a few minor hints after formulating Part A I can handle the question. Thanks for the help!

 

[Ray Vickson]

Homework Statement

[/B]
Suppose the solution set of some system Ax = b , Where A is a 4x3 matrix, is

*Bold characters are vectors*

x_1= 1 + 3t
x_2 = 2 - t
x_3 = 3 + 2t

where t is a parameter and can be any number.

a) How many pivots are in the row echelon form of A?

b) Let u, v, w be the columns of A. Do they span the whole R^4? Explain.

c) Is u, v, w a linearly independent set? Explain.

d) Produce a linear relation for u, v, w. ( Hint: what is the solution of the system Ax = 0 ?)​

Homework Equations

The Attempt at a Solution

My original attempt I tried to solve for the parameter t in hopes that this would give me a coefficient matrix for x that I could substitute in the equation for x but I could not make this work. I also realized that the solution set is linearly dependent as one of the rows can be expressed as linear combinations of the other rows but I don't know if this is significant information for this problem.

My secondary attempt was to consider a system where the solution set in the form listed above was b, then A would be a 4x3 identity matrix. I then multiplied each side of the equation by the 4x3 Identity matrix but multiplying the equation by 1 was not leading me to the next path, and thus now I sit here in utter confusion trying to make sense of this problem and is making me question my confidence in my other solutions on the assignment .

Note: I am having trouble mainly with formulating an expression or method to solve Part A, but I feel with a few minor hints after formulating Part A I can handle the question. Thanks for the help!

Your three equations make no sense. You say that "bold characters are vectors", then write $\mathbf{x}_1 = 1+3t$, etc. You have a vector on the left and a scalar on the right.

 

[Ty Ellison]

Hi, you’re totally right! I didn’t know how use the text editor in P.F. to write a matrix so I attempted to write out the equations. The matrix is given in vector parametric form in the picture I attached. Sorry for the confusion and thanks for the help!

 

[Ray Vickson]

Hi, you’re totally right! I didn’t know how use the text editor in P.F. to write a matrix so I attempted to write out the equations. The matrix is given in vector parametric form in the picture I attached. Sorry for the confusion and thanks for the help!

To get a (column) vector, use either "\pmatrix{..your vector..}" or "\ begin{bmatrix} ... your vector ...\ end{bmatrix}" Your vector has the form "a1 \\ a2 \\ a \\ ..." in either case, so that "\\" ends one row and/or starts another. (Remove the space between the "\" and the word "begin" or "end"; I inserted them just to keep the typsetter from going crazy.) Note that in the first form the whole vector is inside curly brackets "{ }" but not in the second one.
These give:
$$\mathbf{x} = \pmatrix{a_1 \\a_2\\a_3} \; \Leftarrow \; \text{first form}$$
and
$$ \mathbf{x} = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \; \Leftarrow \; \text{second form} $$
The choice is yours.

Right-click on the formula and ask for display math as tex commands to see the actual typed forms.

Alternatively, you could have said that "$x_1, x_2, x_3$ are the three components of the vector $\mathbf{x}$"; then what you wrote would make perfect sense.

 

[Ty Ellison]

To get a (column) vector, use either "\pmatrix{..your vector..}" or "\ begin{bmatrix} ... your vector ...\ end{bmatrix}" Your vector has the form "a1 \\ a2 \\ a \\ ..." in either case, so that "\\" ends one row and/or starts another. (Remove the space between the "\" and the word "begin" or "end"; I inserted them just to keep the typsetter from going crazy.) Note that in the first form the whole vector is inside curly brackets "{ }" but not in the second one.
These give:
$$\mathbf{x} = \pmatrix{a_1 \\a_2\\a_3} \; \Leftarrow \; \text{first form}$$
and
$$ \mathbf{x} = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \; \Leftarrow \; \text{second form} $$
The choice is yours.

Right-click on the formula and ask for display math as tex commands to see the actual typed forms.

Alternatively, you could have said that "$x_1, x_2, x_3$ are the three components of the vector $\mathbf{x}$"; then what you wrote would make perfect sense.

Thank you, I will try retyping the question asap!

 

[SammyS]

Homework Statement

[/B]
Suppose the solution set of some system Ax = b , Where A is a 4x3 matrix, is

*Bold characters are vectors*

x_1= 1 + 3t
x_2 = 2 - t
x_3 = 3 + 2t

where t is a parameter and can be any number.​

...

Below is a snip from the image referenced by your thumbnail. In my opinion the only significant problem with your notation above was to bold face for the components of the x vector.

Now let's look at your second attempt at a solution:

The Attempt at a Solution

My original attempt ...

My secondary attempt was to consider a system where the solution set in the form listed above was b, then A would be a 4x3 identity matrix. I then multiplied each side of the equation by the 4x3 Identity matrix but multiplying the equation by 1 was not leading me to the next path, and thus now I sit here in utter confusion trying to make sense of this problem and is making me question my confidence in my other solutions on the assignment .

You also included this in your image:

Notice that $\ A\, \vec x \$ results from multiplying a 4×3 matrix by a 3×1 vector so your result should have a dimension of 4×1, not 3×1 .