Linear Transformation: Is T(U) a Subspace of R^m?

[kingwinner]

1) True or False? If true, prove it. If false, prove that it is false or give a counterexample.
1a) If a linear transformation T: R^n->R^m is onto and R^n = span{X1,...,Xk}, then R^m = span{T(X1),...,T(Xk)}
1b) If T: R^n->R^m is a linear transformation and U is a subspace of R^n, then T(U) is a subspace of R^m.



2) Let T: R^2->R^4 be a linear transformation induced by the matrix A=
[1 4
2 3
3 2
4 1]
Find a vector X E R^2 such that T(X) is as close as possible to [4 6 6 4]^T



I have an exam tomorrow. These are the past exams questions that I am having terrible trouble with. Can someone help me? I seriously thought about these questions, but still can't come up with any clue...I really want to provide some attempt, but I don't even know how to begin...

Any help/hints is greatly appreciated!

 

[tronter]

1 (a) First what is the definition of onto or surjective? Every point in the domain spans the codomain. Use this fact to answer your question.

 

[radou]

Regarding 1b), simply take two vectors a, b from T(U) and think of a condition which must be satisfied in order for T(U) to be a subspace.

 

[siddharth]

2) Let T: R^2->R^4 be a linear transformation induced by the matrix A=
[1 4
2 3
3 2
4 1]
Find a vector X E R^2 such that T(X) is as close as possible to [4 6 6 4]^T

For this question, you need to find a "best approximation" $$u \in \mathbb{R}^2$$ to
$$b= \left( \begin{array}{c} 4 \\ 6 \\6 \\ 4 \end{array} \right)$$

Have you learned the theorem which says that if u is a best approximation, and A is the matrix of the linear transformation, $$A^T(Au-b)=0$$?

Solve for u to find the best approximation. You could apply QR factorization to A to further simply the solution process.

 

[kingwinner]

For this question, you need to find a "best approximation" $$u \in \mathbb{R}^2$$ to
$$b= \left( \begin{array}{c} 4 \\ 6 \\6 \\ 4 \end{array} \right)$$

Have you learned the theorem which says that if u is a best approximation, and A is the matrix of the linear transformation, $$A^T(Au-b)=0$$?

Solve for u to find the best approximation. You could apply QR factorization to A to further simply the solution process.

Thanks, I have learned this but I have never thought of it...what a great method

 

[kingwinner]

But I am still pretty lost with question 1b...

I know the definition of subspace, but I simply don't know how to apply it in this situation...

U is a subsapce of V iff
1) 0 E U
2) X,Y E U => X+Y E U
and 3) X E U, a E R => aX E U

 

[radou]

But I am still pretty lost with question 1b...

I know the definition of subspace, but I simply don't know how to apply it in this situation...

U is a subsapce of V iff
1) 0 E U
2) X,Y E U => X+Y E U
and 3) X E U, a E R => aX E U

U is a subspace of V if, for every a, b from U, and for every salars x, y, xa+yb is in U. Use that fact.