Let F be the field with two elements, F = {0, 1}. How many injective linear mappings are there A: F^2 -> F^3 with the property that A(1,0) -> (1, 1, 1)?

Find a basis for R2 that includes the vector (2,2).

Let Rz, be a linear space with all polynomials p(z) = ax² + bx+c(a,b,c € R) whose degree is no more than 2. Let L be the operator on R[r], defined by 4 Lar²+ bx+c)=(a + 3b+6c)x²+(-a-3b-4c)x+(a+b+ 3 (1) Find the matrix A representing with respect to basis [1, 2, 3x²]; 2 3 (2) Find the matrix B representing with respect to basis (1-2r+32²,-1+ 2r, -1+3z²); (3) Is the matrix A diagonalizable? Why?

6. P2 denotes the set of all polynomials of degree ≤ 2, in a single variable x, with real number coefficients. P2 is a vector space under the usual operations of polynomial addition (combining like terms) and scalar multiplication. Determine whether each set of polynomials in P2 is linearly independent or linearly dependent. If the set is linearly dependent, find a dependence relation. If the set is linearly independent, demonstrate that any dependence relation has to be trivial. (a) {2x, x+3} (b) {x, 2x+5, 10} (c) {x² + 5x+1, x²-7x, x² +10, 2x² + 3x-4} 1ST (d) Make a linearly independent set of three polynomials in P2. Explain how you know.

Find an algebraic description of the range of A by row reducing A^T and using the relation between R(A) and Row(A^T ) in order to obtain a basis of R(A).

Suppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.

Determine whether the function defined by = 2x1y1 − X1Y2 − X2y1 + 2x2y2 - - is an inner product on R².

Suppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect.  Please Provide Unique Answer. Thank you!

Let T: R² →R² be a linear transformation that maps u = ·[³][:]~~-~~-~-:-(¯;}]} and maps v= The image of 3u is 2 5 into 6 1 3 3 into 1 3 Use the fact that T is linear to find the images under T of 3u, 4v, and 3u + 4v.

Suppose T: R^3 —> R^3 is a linear transformation and (v1,v2,v3) is a linearly dependent set of vectors in R^3. Prove that {T(v1),T(v2),T(v3)} must also be a linearly dependent set.