# Thread: Linear Algebra: Linear Transformations

1. ## Linear Algebra: Linear Transformations

I am working on several linear algebra problems and found myself stuck on two problems. Any helpful hints or suggestions on how to go about solving them will begreatly appreciated.

1) Given a 3 x 5 matrix and T(x)=Ax. Find range(T).
How do I find the range?

2) Let V be an inner product spae with a subspace W having B={w1,w2,...,wn} as an orthonormal basis. Show that the T(v)= the sum i=1 to n of <v,wi>wi is a linear transformation.

2. Originally Posted by mathwiz2006
1) Given a 3 x 5 matrix and T(x)=Ax. Find range(T).
How do I find the range?
It depends on the matrix A.

2) Let V be an inner product spae with a subspace W having B={w1,w2,...,wn} as an orthonormal basis. Show that the T(v)= the sum i=1 to n of <v,wi>wi is a linear transformation.
Remember you need to show that $\displaystyle T(k\bold{x}) = kT(\bold{x})$ and $\displaystyle T(\bold{x}+\bold{y}) = T(\bold{x}) + T(\bold{y})$.
Can you show that?

3. 1) A= [3,-2,6,-1,15; 4,3,8,10,-14; 2,-5,1,-4,20]

2) I know that I must show
and .
but I am mostly confused because it is the sum of the given and am having trouble showing that they work