In F^n let e_j denote the vector whose jth coordinate is 1 and whos other coodinates are 0 prove that generates F^n […]

A matrix M is called skew-symmetric if M^t=-M clearly a skew symetric matrix is square. Leet F be a field prove that the set W_1 of all skew symmetric nxn matrices with entries form Fis a subspace […]

Let W1 and W2 be subsapce of a vector space V. Prove that W1 union W2 is a. subspace of V if and only if W1 subset W2 […]

For each of the following maps. Determine if it is a linear transformation if it is, find its standard matrix. If it is not explain which property or linear tranformation it violates […]

Show that if F=R the x=axis and y-axis are subspaces of R2 plane but th union of these two axis is not a subspace […]

Let S = bee a subset of the vector space F3 prove that if F=R the S in linearly independent. Prove that if F has characteristic 2, then S is linearly dependent […]

In. Mmxn(f). Leet Ein denote the matrix whose only nonzeroe entry it the ith ow and jth colunm. Prove that is linearly independent […]

The vectors u_1, u_2, u-3 , u_4 and u_5 generarte R3 find a subset of the set that is a basis for R3 […]

Let S = { u_1, u_2, … , u_n} be a linearly independent subset of a vector space V over the field Z_2. How many vectors are there in span(S)? […]

Let u and v be distinct vectors in a vector space V. Show that {u,v} is linearly dependent if and only if u and v is a multiple of the other. […]

Let R R2 to R2 be the linear transformation given by R = Ax and S be R2 to R2 be the tranformation give by S=Bx where A and B find the stand matrix of the linear transformation […]

Let A and B be arbitray 2×3 matrix explain why every column of the 3×3 matrix C must belong to the plane in R3 […]