Test 2.


Problem 1. Prove that the range of any linear transformation from a vector space V to a vector space W is a subspace of W.

Problem 2. Consider the following addition and scalar multiplication on the set of all 2 by 2 matrices:

[ a b ]
[ c d ]
+
[ e f ]
[ g h ]
=
[ a+f b+e ]
[ c+h d+g ]
k
[ a b ]
[ c d ]
=
[ ka kb ]
[ kc kd ]

Is the set of 2 by 2 matrices with these operations a vector space?

Problem 3. Find the standard matrix of the linear transformation of R2 which first turns a vector 30 degrees counterclockwise and then reflects it about the line y=3x.

Problem 4. Find the kernel of the linear transformation from R4 to R2 with the following standard matrix:

[ 1 2 4 5 ]
[ 1 3 4 0 ]

Problem 5. Is it true that the column space of the above matrix is the whole R2?

Problem 6. Find the kernel and the range of the linear transformation from C[0,1] to R which takes every function f(x) from C[0,1] to the number f(1)+f(0).