- 2 -1 -5 2 1 - 9 3. Let T= 2 -8 - 2 1 Prove that span - 2 1 and | | 3 11 5 -5 -13 -7 | 3 span 1 - 4 are both invariant subspaces under T. - 4 N 3

Q: (2) Find a basis for the subspace W = {(a +c, 2a – b, 3a – 26 – c,b+ 2c) | a, b, c € R} of R.

A: We need to find the basis for the given subspace:

Q: The set of all polynomials of the form ax³ bx² cx + 2, where a,b and can real numbers is a subspace…

A:

Q: 5 9. and define p : R² → R² by p(x) = Ax (that is, p = PA). Find Let A -1 -1 all 9-invariant…

A:

Q: -5 5. 10 8. Let A= 4 -7 -11, and define W = {C E M2x3 CA = 02x3}- %3D %3D -2 7. a. Prove that W is a…

A:

Q: -5 -4 -2 and -6 -5 , find the closest point to v in the subspace W spanned by 8 Given v = -2 -6 -8…

A:

Q: 2.14. Prove that two subspaces V1 and V2 are linearly independent if and only if Vịn V2 = {0}.

A: V1 and V2 are two subspace.

Q: a Let M = . -b a,b,c e R},N x, y e R} be two subspaces of M2x2. Find the basis for M, N and MON.

A: We will find the bases of M and N. Then we will find the intersection of their basis vectors, which…

Q: -9 4 -3 -21 Find the vector in the subspace W spanned by which is closest to –1 -3 -5 Answer:

A:

Q: 1 - 2 - 3 Let v1 = V2 - 4 and w = - 1 Determine if w is in the subspace of R° generated by V1 and…

A: Span of vectors

Q: 4. Let S be the subspace of R4 spanned by x = (1,0,-2,1)7 and x2 = (0,1,3, -2)T. Find a basis for S

A: Consider a matrix A as follows.

Q: Let 1 -3 and let W the subspace of R' spanned by u and v. Find a basis of W-, the orthogonal…

A:

Q: 10 Find the projection of 69 -15 -4 subspace W of R³ spanned by -6 and - 14 onto the √] 6 -

A: Given v→=10-14-15 The objective is to find the projW(v→).

Q: 3.12. Show that for a subspace E we have (E-)- = E. Hint: It is easy to see that E is orthogonal to…

A:

Q: 4. Find a basis and the dimension of the subspace W of V = M22(R) spanned by %3D -5 2 -4 A = B = C =…

A: Given:          V = M2 × 2(ℝ) spanned by A = 1-5-42 ; B = 11-15 ; C = 2-4-57 and D = 1-7-51   To…

Q: 3 5 2 and find the closest point to i in the subspace W spanned by -5 Given i -6 3

A: Givenv→=34-55and let s1→=52-61 and s2→=0734The formula of orthogonal projection of v onto span of s1…

Q: -11 -3 1 2 Find the orthogonal projection of v = onto the subspace W spanned by 43 -5 -3 3 4 5 projw…

A: Given: v→=-11243-3 and W is a subspace spanned by -32-53, -3254, 12-35.

Q: Use Theorem 4.2.1 to determine whether the set of all vectors of the form (0,0,m) form a subspace of…

A: Option first is correct

Q: 2 7 -6 find the closest point to v in the subspace W spanned by -4 9. Given v = -2 and 2 -9 -18

A:

Q: 4. Let H be the set of 2 × 2 whose trace is equal to zero. That is [a11 a12 Н- E M2(R) | a11 + a22 =…

A:

Q: Find the orthogonal projection of -19 5 -5 -12 onto the subspace W of IRª spanned by 1 -4 and 5 10

A: As per the orthogonal Decomposition  theorem  Let us assume W is a subspace of Rn then each y in Rn…

Q: Which of the following is a basis for the subspace {(: {{:::):} {(:::)(:} a b c W = a = b+c and a,…

A:

Q: 3. This problem is about set addition and subspaces. Let d and d2 , and V = span{di}, U = span{d2}.…

A:

Q: -5 -2 -4 Given v = find the closest point to v in the subspace W spanned by -2 and 7 70

A: Let consider  v=−19−2−3, u1=−5−26−1, u2=−4−4770

Q: (2) Find a basis for the subspace W%3D { (а+c, 2a — b, За — 2b — с, b + 2с) | а, b, с € R} of R4.

A:

Q: 8) Describe geometrically all subspaces of R" for n 1, 2, 3.

A: We will do this for all three parts; n = 1, 2, and 3 (in as clear handwriting as possible)

Q: Determine whether the following are subspaces of R2×2: The set of all symmetric 2 × 2 matrices

A:

Q: Find a basis for the subspace and state the dimension. p– 2q 2р-5r :P,q,r e R а. -2q +2r -Зр+ 6r| 1…

A:

Q: 4. Let H be the set of 2 x 2 whose trace is equal to zero. That is a11 a12 Н- E M2(R) | a11 + a22 =…

A:

Q: 5. Let S = {x + 1,x2 – 2, x – 1,3}. Find a basis for the subspace W that is spanned by S.

A:

Q: 6. a. Give a basis for the orthogonal complement of the subspace V C R' given by V = {x € R* : xị…

A:

Q: Example (3.8): Let X = C[0,1] and let W = [fo, fu .] such that fi(x)=x', we observe that W is an…

A:

Q: 1 4 Let 3= {b1, b} be a basis of a subspace H or R3, where b, = 5 1. b, = -7 If x = 4 is a vector in…

A:

Q: Let A = (; ). Find the basis for each of the four subspaces associated with A. 2 5 8

A:

Q: Consider the subspaces U = span{[1 -4 -5],[-1 –6 -13]} and W = span{[4 -4 4],[-2 0 -2]} of V = R'x3.…

A: Any vector belongs to a space spanned by v1 and v2 is the linear combination of v1 and v2.

Q: Consider the subspaces U = span{[3 -3 1],[12 3 -7]} and W = span{[-3 2 0],[1 8 -6]} of V = R1x3.…

A:

Q: 1 -2 -3 Let v, ,V2 - 7 and w -3 Determine if w is in the subspace of R generated by v, and v2. -6 11…

A: To determine  If W is in tg subspace of R^3 generated by v1 and v2.

Q: Which of the following statements are/is TRUE? 6 1 (1) 5 3 is in the subspace of 2 x 2 matrices…

A:

Q: 12- Let X=C[0,1] and let W=[fo,f1,...] such that f;(x) = x', then we have W is A-an infinite…

A:

Q: Consider the subspaces U = span{[3 5 0],[0 12 -6]} and W = span{[–3 2 -1],[-15 7 -11]} of V = RIX3.…

A:

Q: Let fi = 1+2r + 3x², f2 = 2 + 3x + 4x². B = (f1, f2) is a basis for the subspace V = {ao + a1x +…

A: For the given problem on vector spaces we need to determine the coefficient s .

Q: -15] 1 [2] Find the orthogonal projection of v: -11 onto the subspace V of IR spanned by and 0 -2…

A:

Q: Consider the subspaces U = span{[3 -4 -2],[6 -8 2]} and W = span{[ -2 -2 3],[2 -12 14]} of V = R'x3…

A:

Q: Show that W={(x1,x2,x3,x4) : x4-x3=x2-x1} is a subspace of R^4  spanned by vectors…

A:

Q: -3 -10 -2 -2 and let W the subspace of R4 spanned by ű and v. Find a basis of W+, the orthogonal…

A:

Q: Let H span {(-13, –1, -3),{-9,2, –2), (-1, –5, 0)}. A basis for the subspace H C R³ is { }. vector…

A:

Q: 6. (a) Determine whether the set of all vectors of the form (a,b,c), where b = a+c, is a subspace of…

A: To show the subset W is a subspace , we need to show that (a) 0∈W (b) For every w1 , w2∈W , we have…

Q: [x++2x2 -X1 :X1, X2 real numbers is a subspace of R4 with dimension dim H H = 3x1-X2 X1+X2

A: I jave used that dimension of a space is equal to cardinality of basis And found a basis of H which…

Q: The sequence B = (3,5) is a basis for the subspace () -- V = { 22: 2xi + x2 – 13 = 0}. The vector w…

A:

Q: 12- Let X=C[0,1] and let W=[fo,f1...] such that f;(x) = x', then we have W is A-an infinite subspace…

A:

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.