6. (15 points) Let B = (2, 1, 1). Prove that B is linearly independent and...
1· Let S {u,v) be a linearly, independent set. Prove that(u+ v.u-v) is linearly independent. 2. Let H :2y1. Prove that H is not a subspace of f2.
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 –...
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...
Let A be a 5 x 3 matrix whose columns are linearly independent. Prove: If B is an invertible 3 x 3 matrix, then the columns of AB are linearly independent. Let A be a 5 x 3 matrix whose columns are linearly independent. Prove: If B is an invertible 3 x 3 matrix, then the columns of AB are linearly independent.
Proble m 3. Let T: V ->W be (1) Prove that if T is then T(),... ,T(Fm)} is a linearly indepen dent subset of W (2) Prove that if the image of any linearly in depen dent subset of V is linearly indepen dent then T is injective (3) Suppose that {,... ,b,b^1,...,5} is Prove that T(b1), .. . , T(b,)} is a basis of im(T) (4) Let v1,. Vk} be T(v1),..,T(vk) span W lin ear transform ation between vector...
15) 9. I u.v is linearly independent, prove that 2ï.v-i is also linearly independent.
Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the columns of B are linearly independent. Let A be an 3 x 4 matrix, and B an 4 x 3 matrix. Prove: If AB Is, then the columns of B are linearly independent.
3. Which of the following set of vectors in R3 are linearly independent? (a) (6, -11, 2); (-6, 13, -2), (b) (2,6,6); (2,7,6); (2,7,7), (c) (1,-1,3); (-2,0,5); (3,-1, 1); (2,2,3). Explain your answer. Which of these systems forms a basis in R3.
2. (a) Is the collection 1 2 linearly dependent or linearly independent? Justify. [2 (b)Describe (geometrically) the following span as lines, planes or all of R3 2 Span〉 | 2 -3
Let T:V→WT:V→W be an isomorphism. Prove that if {w⃗ 1,w⃗ 2,…,w⃗ n}{w→1,w→2,…,w→n} is a linearly independent set in WW, then the preimages of {w⃗ 1,w⃗ 2,…,w⃗ n}{w→1,w→2,…,w→n} is a linearly independent set in VV.