30. In the vector space R2x2, 1 A = 0. 1 0 B = C = -1 -3 1 are given. (a) Is {A, B, C} linearly independent? (b) Use the standard inner product in R2x2 to find an orthogonal basis for the subspace W =< A, B,C>. 1. (c) Which matrix in (A, B,C) is closest to

Q: Find an invertible matrix Q and a matrix J in Jordan canonical form such that Q-'AQ = J, where 7 1 0…

A: Given matrix is A=710-150106 To determine matrix Q and J

Q: 2. Find a vector orthogonal to the nullspace of the matrix [1 27 6. 4 L5 -1 2.

A: According to the given information it is required to find a vector orthogonal to the null space of a…

Q: In the vector space P, of polynomials of degree atmost two, the first column of the matrix of…

A: The first column of the transition matrix obtained from the expression of first vector a1 with…

Q: 3. In the of square matrix M2x2 find representation of vector A = from the L4 space basis U2 %3D ;…

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Q: 2. Find the matrix of the linear transform æ → Ax = (x1-82 + x3, x1 – #2, x2 – 23), x = (x1, #2, 13)…

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Q: In the vector space P,, of polynomials of degree at most two, the matrix of transition from the…

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Q: 5.3. Prove that a linearly independent system of vectors v1, V2, . .. , Vn in a vector space V is a…

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Q: §5.1, Exercise 17. For which scalars a, b, c do the solutions to the equation ax + by = c form a…

A: We have to find for which scalars a,b,c do the solutions to the equation ax+by=c from a  subspace of…

Q: Q2) Let M2x2 be the vector space of all 2X2 matrices, and defined T:M2x2 → M2x2 , such that T(A)=…

A: T:M2×2→M2×2, such thatTA=A+AT

Q: 4. Present the standard basis vectors e, and ez in R? as linear combinations of (3, 1) and (1, 3).

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Q: [1 Let matric A be a square hermitian matrix= |2 4 5 5 1. L3 Use Gram-Schmidt process to find the…

A: Consider the given matrix, A=123235351 Let u1=123, u2=231, u3=351 Step 1: u1=v1 v1=123 Step 2:…

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Q: Solve: (a) If a 7 x 9 matrix A has rank 5, what are the dimensions of the four fundamental…

A: If the order of the matrix is m×n and rank of the matrix is r. Then dimension of column space =r…

Q: 2) Find all the invariant subspaces of the following matrix 0. 1 0 1 1 1

A: Given matrix is, Let, A=010010111 Now, calculating the eigen value of the given matrix.…

Q: 6. Consider the 3-dimensional vectors El a = 2 and Find a matrix P for which, for each 3-dimensional…

A: 6) Given two vectors are a1=122 and a2=102 .

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Q: 4. Generate a matrix p using the basis of R³ IS the matpix usual P is linearly independent ? Find…

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Q: §8.3, Exercise 4. Let E¡j be the 2 × 3 matrix whose entry in the ith row and jth column is 1 and all…

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Q: 5.2. Prove that if V is a vector space having dimension n, then a system of vectors V1, V2, ..., Vn…

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Q: 2. Consider the quadratic form Q : R° → R, where Q(x1, 22, x3) = –2x – 8x – 18x3+ 8x1x2 + 4x1x3 +…

A: The given quadratic form is Q(x1,x2,x3)=-2x12-8x22-18x32+8x1x2+4x1x3+4x2x3 The quadratic homogenous…

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A: Given that,   Let x1,...,xk be linearly independent vectors in Rn, and let A be a nonsingular n×n…

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Q: Suppose there is a basis a of R2, for which the system |[T]-AI|=0 has one, and only one, real…

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Q: 1. Let M,(R) be the vector space of n x n matrices with real entries and let {A € M3(R) | Aª = -A}.…

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Q: 4. Consider the vector space P, and the bases S = {1,x,x²} S' = {3,x-1,x² - 2x}. Find the transition…

A: We can solve this as follows:

Q: 2. Q is an xn symmetric matrix, b is a vector of size n and c is a scalar If f(x) = x"Qx + b"x + c:…

A: given  fx=12xTQx+bTx+c it is known that  gradient of the function can be expressed as…

Q: Given a basis B = {V1, V2,, Vn} of Subspace S, there is a unique way to express any ... vE S as a s…

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Q: Suppose masses m1, m2, m3, m4 are located at positions x1, x2, x3, x4 in a line and connected by…

A: (a) The 4 x 4 matrix equation relating the column vectors f and x will be:

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Q: Suppose V and W are finite-dimensional and T E L(V,W). Show that with respect to each choice of…

A: Suppose V and W are finite-dimensional and T∈LV,W. To show: to each choice of bases of V and W, the…

Q: (2) Give an example of each of the following: (a) A matrix Q that has orthonormal columns but QQT +…

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Q: In the vector space R2 consider the two bases given by [u1, u2] with u1 = (1, 2)T, u2 = (2, 5)T and…

A: To  find the transition matrix Given, u1=12 , u2=25 , v1=32 , v2=43  The transition matrix of u1,u2…

Q: 1. For each matrix A, find the singular value decomposition in the matrix form A = UEVT. 11 8 3 7 1…

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Q: Let A be an 2 x 3 matrix and b be a vector in R" such that the system Ax = b has solution set 0. 0.…

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Q: Please see image

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Q: S. The image of the vector i + j – 2k under a mapping (1 2 3) defined by the matrix M, where M =0 1…

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Q: If By = {v1, v2, ..., Vn} is a basis for V, show that there exist n +1 nested subspaces of V.

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Q: 2. Find a 3 x 4 matrix R for which the null space of R is spanned by the vectors -- -E 0. v = u =…

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Q: Suppose a matrix A has singular value decomposition A = UEVT, 0 0 300 0 2 0 0 0 1 where U = 1 0. 0.…

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Q: =,4} is linearly dependent. 2, V3). 3 000- 2 3 5. undant vector of R³ (b) {1, ₂} is linearly indepe…

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Q: In the vector space P., of polynomials of degree at most two, the second column of the matrix of…

A: Given : vector space P2, of polynomials of degree at most two To find : the second column of the…

Q: d) Consider the linear map T: C→ C whose matrix in the standard basis e1, e2, e3 is (0 1 0V 0 0 1 0…

A: Given linear map T:ℂ3→ℂ3 whose matrix in the standard basis e1,e2,e3 is 010001000. Then, the linear…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.