Linear Algebra - Basis and Kernel

[succubus]

Consider a 5 x 4 matrix...

We are told that the vector,

1
2
3
4
is in the kernel of A. Write
v4

as a linear combination of

v1,v2,v3I'm a bit confused. Since this is a kernel of A, the kernel is a subset of R^m, therefore the other columns are linear combinations and therefore redundant. (since this is the only column represented) So, that means I can have the columns be anything I want, so why can't they just all be the same? Is this too easy or am I missing something?

 

[Office_Shredder]

What are the v_i here?

 

[succubus]

The v_{4} seems like

1
2
3
4
​

since the problem is asking for v_{4} to be written as a linear combination of \overline{v_{1}},\overline{v_{2}},\overline{v_{3}}

 

[Mark44]

The v_{4} seems like

1
2
3
4
​

What do you mean that v4 "seems like" the above? It either is or isn't a vector with coordinates 1, 2, 3, and 4.

since the problem is asking for v_{4} to be written as a linear combination of \overline{v_{1}},\overline{v_{2}},\overline{v_{3}}

I'm guessing that v1, v2, v3, and v4 are the columns of your matrix.

What does it mean that a vector x is in the nullspace of a matrix?
If you carry out the multiplication Ax, what do you get?
If you carry out the same multiplication, but using the vectors v1, v2, v3, and v4, what do you get?

 

[succubus]

Since this is a column of only 1 vector representing a kernel, would this represent a plane? I have the equation as
\overline{v_{1}} + 2\overline{v_{2}} + 3\overline{v_{3}} + 4\overline{v_{4}} = 0


First, I solve for \overline{v_{4}}

So is it

\overline{v_{4}} = c1(4\overline{v_{4}}) + c2(2\overline{v_{4}}) + c3(3/4\overline{v_{4}})

Referring to above poster.

When I read the question, it basically seemed confusing at first. I was thinking of it in terms of a redundant column vector. Now I'm thinking each number is the coefficient to the column vector and it represents a plane. Am I correct in assuming this based on the information?

 

[Office_Shredder]

Can you post your question, precisely as it is stated, with no shorthand use of pronouns? Just straight from the text.

 

[succubus]

I thought I had it exact, but my formatting is all wrong. I kept copying and pasting and realized that I ws calling v's x's. I fixed it. :/ Sorry for the confusion.

 

[Mark44]

Since this is a column of only 1 vector representing a kernel, would this represent a plane?

The problem states that [1 2 3 4] is in the kernel, not that it represents the kernel. We know that the kernel is at least 1-dimensional (a line through the origin). It might be that the kernel is 2-dimensional (a plane in 4D) or higher.

I have the equation as
\overline{v_{1}} + 2\overline{v_{2}} + 3\overline{v_{3}} + 4\overline{v_{4}} = 0


First, I solve for \overline{v_{4}}

So is it

\overline{v_{4}} = c1(4\overline{v_{4}}) + c2(2\overline{v_{4}}) + c3(3/4\overline{v_{4}})

No, but I think you have the right idea, assuming that my interpretation of this problem was correct; namely, that v1, v2, v3, and v4 are the columns of A.. Try again, and be more careful with your subscripts.

Referring to above poster.

When I read the question, it basically seemed confusing at first. I was thinking of it in terms of a redundant column vector. Now I'm thinking each number is the coefficient to the column vector and it represents a plane. Am I correct in assuming this based on the information?