(6 pts) 3. Prove whether or not the set of ordered pairs of the form (w...
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
suppose that s=(v1,v2,......vm) is a finite set of linearly independent vectors in V, and w ∈ V some other vector. Let T= S ∪ (W). Prove that T is not linearly independent if and only if w∈ span(s).
1. Determine whether the following set is linearly independent or not. Prove your clas a. [1+1, 2+2-2,1 +32"} b. {2+1, 3x +3',-6 +2"} 8. Let T be a linear transformation from a vector space V to W over R. . Let .. . be linearly independent vectors of V. Prove that if T is one to one, prove that (un)....(...) are linearly independent. (m) is ) be a spanning set of V. Prove that it is onto, then Tu... h...
1) Determine if w is in the subspace spanned by v1, v2, v3 2) Are the vectors v1, v2, v3 linearly dependent or independent? justify your answer Question 2. (15 pts) Let vi=(-3 0 6)", v2= (-2 2 3]", V3= (0 - 6 37, and w= [1 11 9". (1). Determine if w is in the subspace spanned by V1, V2, V3. (2). Are the vectors V1, V2, V3 linearly dependent or independent? Justify your answer
Question 2. (15 pts) Let vi= [-3 0 6)". Vy=[-2 2 3". Vg= [0 - 6 3), and w=[1 14 97 (1). Determine if w is in the subspace spanned by V. V2 V3. (2). Are the vectors Vi, V2, V3 linearly dependent or independent? Justify your answer.
s=3 Let sor,+r,+r, = . Determine whether the set 2-X.SX-X?.6-(s+1)x+x' in P, is early independent or linearly dependent. If the set is linearly dependent then write one of the tors as a linear combination of the other two vectors in the set.
Please give answer with the details. Thanks a lot! Let T: V-W be a linear transformation between vector spaces V and W (1) Prove that if T is injective (one-to-one) and {vi,.. ., vm) is a linearly independent subset of V the n {T(6),…,T(ền)} is a linearly independent subset of W (2) Prove that if the image of any linearly independent subset of V is linearly independent then Tis injective. (3) Suppose that {b1,... bkbk+1,. . . ,b,) is a...
1. (15 points) Prove whether the following sets are linearly dependent or independent, and determine whether they form a basis of the vector space to which they belong. s 10110 -1 ) / -1 2) / 2 1 17 ) } in M2x2(R). "11-21 )'(1 1)'( 10 )'(2 –2 )S (b) {23 – X, 2x2 +4, -2x3 + 3x2 + 2x +6} in P3(R) (the set of polynomials of degree less than 3. (c) {æ4—23+5x2–8x+6, – x4+x2–5x2 +5x-3, x4+3x2 –...
Question 2. (15 pts) Let Vi= (-3 0 6)", v2= (-2 2 3)", V3= [0 - 6 3)", and w= [1 14 9)? (1). Determine if w is in the subspace spanned by V1, V2, V3. (2). Are the vectors Vi, V2, V3 linearly dependent or independent? Justify your answer.
Question 2. (15 pts) Let vi= (-3 0 6)", V2= (-2 2 317, V3= [0 - 6 3)", and w=(1 14 9) (1). Determine if w is in the subspace spanned by va, V2, V3. (2). Are the vectors V1, V2, V3 linearly dependent or independent? Justify your answer.