Let A be a 3 × 3 matrix and let x1, x2, x3 be vectors in R3. Show that if the vectors y1 = Ax1, y2 = Ax2, y3 = Ax3 are linearly independent, then the matrix A must be nonsingular and the vectors x1, x2, and x3 must be linearly independent.

Q: Prove that if B is the matrix obtained by interchanging the rows of a 2 ×2 matrix A, then det(B) =…

A: Let the matrix A and the new matrix B be as follows.

Q: If C is a real square matrix, then C² + I (I is the identity matrix ) is non-singular. - The vector…

A: Find the dimensions of U and V.  Then use the result if two vector spaces have same dimension then…

Q: Show that if the entries in each row of an n × n matrix A add up to zero, then the column vectors of…

A: linearly dependent means det of A is zero

Q: Let A be an m×n matrix. Show that if A has linearly independent column vectors, then N(A) = {0}.

A:

Q: Let x and y be vectors in Rn and let Q be an n × n orthogonal matrix. Show that if z = Qx and w = Qy…

A:

Q: Let A be an m×n matrix and let B be an n×p matrix. Show that if the columns of B are linearly​…

A:

Q: Construct a 3x3 matrix A, with nonzero entries, and a vector b in R such that b is not in the set…

A: A transformation is called linearly independent when the nullity of the transformation is 0.…

Q: Let x1, x2, and x3 be linearly independent vectors in R4 and let A be a nonsingular 4 × 4 matrix.…

A: Given that:

Q: For the matrix A in , use the Cayley ­Hamilton Theorem to compute A2, A3, and A4 by expressing each…

A: Given matrix is

Q: Let A be an m×n matrix. Explain why the followingare true. Any vector b ∈ Rm can be uniquely written…

A: For  matrix A, N(A) is the null space of A. R(A) is the column space of the matrix A and R(AT) is…

Q: Show that A + AT is symmetric for any square matrix A.

A:

Q: Find a matrix A such that W = Col A. W [2s-5t] 2t 2s+t :stin R

A:

Q: Let A be an (n x n) matrix and za column vector. If AA = I then Ax|| = ||r||. %3D %3D O True O False

A:

Q: let Q(x)= x12 -8x1x2 -5x22 , then matrix A of the quadratic from is A=?

A: Explanation is given below..

Q: Let A be a 2 × 2 matrix with linearly independentcolumn vectors a1 and a2. If a1 and a2 are usedto…

A:

Q: Show that if A is an orthogonal matrix, then A−1 is orthogonal.

A: Explanation: Given that,  A is an orthogonal matrix.

Q: Let ~v and ~w be column vectors in R^n, and let A be an n×n matrix. Prove that, if {A (~v), A (~w)}…

A:

Q: Construct a 3x3 matrix A, with nonzero entries, and a vector b in R such that b is not in the set…

A: using the concept of vector space , approches to check one by one option.

Q: Consider the matrix. P = in order 4, where X1, X2, X3 and X4 are the matrix x4 rows of P. If we have…

A:

Q: Show that if A is an n x n matrix, then A" can be written as a linear combination of the matrices I,…

A: Cayley-Hamilton Theorem:- Every square matrix satisfies its own characteristic equation.

Q: Let D = P'AP be a diagonal matrix, wher %3D Pesell thet if D:

A: Given, D=P-1AP be a diagonal matrix, where P is an orthogonal matrix. To prove that A is symmetric.

Q: Prove that if a matrix B has a left inverse then the columns of B are linearly independent.

A: B has a left inverse, that is, there exists an n × m matrix P such that PB = I.

Q: Let A be an m × n matrix. Prove that if the reduced echelon form of A possesses precisely one pivot,…

A: Given that A be an m×n matrix. Suppose that reduced echelon form of A possesses precisely one pivot.…

Q: 4. Determine if the vector b is in the span of the columns of the matrix A, where [1 2 3 A = 7 8 10…

A:

Q: Let A be an n × n matrix and let x and y be vectors in Rn. Show that if Ax = Ay and x /= y, then the…

A:

Q: If A and B are n x n matrices, and vector v is in the image of both A and B, then i must be in the…

A: True or false.

Q: Let A be an m×n matrix. Explain why the following are true. Any vector x in Rn can be uniquely…

A: Given : A is an m×n matrix. To show : a) Any vector x in ℝn can be uniquely written as a sum y+z,…

Q: If v, w E Rª are linearly independent vectors, then there is an invertible matrix P such that Pv and…

A:

Q: X = 4 0 1 1 0 2 -3 6 -12 10 -1 5. - = 1 2 [X1 X2 X3 X4]

A:

Q: Determine if the vector u is in the column space of matrix A and whether it is in the null space of…

A:

Q: X = 4 0 1 0 -3 6 1 2 -12 10 -1 = [X₁ X₂ X3 X4] 5

A: Linear algebra

Q: Let A = UEVT be a SVD of a square matrix A. What is the product of the diagonal elements of E, in…

A:

Q: Let r1, .. r,be vectors in R", and let Q be an m xn matrix. Write the matrix Qr, Qr, as a product of…

A: Given:Let r1.......rp be vectors in ℝnLet Q be an m×n matrix Qr1.......QrpR=r1.......rpQ=Q…

Q: Use the Cayley-Hamilton theorem to find A1, A°, and A“ for the given matrix A. 1 5 1 A = 0 2 1 0 0 2

A:

Q: Show that if A is an n x n symmetric matrix, then |(Ax)•y = x•(Ay) for all x, y in R".

A: In symmetric matrix, transpose of the matrix is equal to the matrix. AT=A

Q: Let A ∈ Rm×n, B ∈ Rn×r, and C = AB. Show that (a) if A and B both have linearly independent column…

A:

Q: For a matrix A, define Col(A) to be the set consisting of all linear combinations of column vectors…

A:

Q: Let A and B be n × n matrices and assume that columns of B are linearly dependent. Show that columns…

A:

Q: Suppose that A is a square matrix and that ∥Au∥ = 1 for all unit vectors u. Show that A is…

A: Given: To show that A is orthogonal, assuming that A is a square matrix. Here Au =1for all unit…

Q: Let A be an n x n matrix. If vi, v2,..., v are vectors in R" such that {Avi, Au2,..., Avn} is…

A:

Q: Let A and B be n x n matrices and c be a scalar. Prove that, if B is obtained by multiplying a row…

A:

Q: Let A = (a;;)ij be an m x n matrix of real numbers, and let f : R" Rm be defined by 0-0 x2 x2 f = A

A: To find the derivative of f given: matrix A which has m rows and n columns  also the function f…

Q: Please see image

A: Given matrices are

Q: If u is a nonzero vector in R", then uu is a rank one matrix. O True O False

A: We have given u is a non-zero vector in ℝn. Then, u=a1a2..an where, at least one  ai≠0 and 1≤i≤n.…

Q: let  A and  B be  square  matrices of order n  such that  Ax=Bx  and x belongs to Rn  then   find…

A: Ax = Bx implies  Ax - Bx = O So, (A - B)x = O where x belongs to Rn  So, x belongs to the null space…

Q: Find all a and B >0 so that the matrix B = 23 6 2 is diagonally dominant;

A: A square matrix is diagonally dominant if   for .

Q: Show that for matrices A, B E RXn, a vector w = ABx can be written as w = E,(b{ x)a;, %3D :D1 where…

A: Since, A,B∈ℝn×n. ∴ A, B have 'n' rows and 'n' columns with entries as real numbers. The given vector…

Q: Find a matrix A such that the given set W of vectors is the column space -r+ 3t r - 2s + 2t -3r+2s…

A:

Q: Let A be a 2x2 matrix, c a scalar. Then det(cA)= c*det(A). Select one: O True False

A: Select one: True or false:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.