D) How do you know that T: P3> P2? (How do you know that the codomain is correct?) E) Find T(x + 2x? + x + 4) F) Find, if possible, a basis for ker(T), and state the nullity of T.

Q: Find a basis of ker T. T: M22 → 1 M22; T( x ) = AX, where A = 0 0 O[1 0 0 1 0 1 1 0 1 0 1 0 0 -1 0…

A: The transformation T:M22→M22; T(x)=AX, where A=1100. We have to find the basis of kerT.

Q: Is B a basis for P2? B = {-x +1 ,2x +3, x² }

A: Given B =  -x+1 , 2x+3 , x2  We know that the standard basis of P2 is   1 , x , x2 .

Q: 4. Apply the principal axis transformation to the curve of second order -x2 + 6xy – y? – 22.x+ 18y –…

A:

Q: 7. Prove that p, (t) = 1+t², p2(t) = t – 3t2, p3(t) = 1 +t – 3t, is a basis for P2.

A:

Q: Let 9. A = 3 9 4 6. -3 2 -2 Find an orthonormal basis of the kernel of A.

A:

Q: -2 Let y = what are the expansion coefficients of y in the basis {vo, V1, V2 , v3 } you found in the…

A:

Q: Explain why S is not a basis for P2. S = {1 – x, 1 – x², -1 – 3x + 4x²} O is linearly dependent. O s…

A:

Q: Explain why S is not a basis for P2.S = {2, x, 3 + x, 3x2}

A: From the given value for S to find the values for pi(x).

Q: Let y = 2 1 what are the expansion coefficients of y in the basis {vo, V1, v2 , v3 } you found in…

A: In the Previous question, the vectors v(3) that is forms an orthonormal basis in R4, The vector are…

Q: Suppose A E R4×3 has the following QR factorization A QR V2 -V2 [1 0 1 0 0 0 0 -1 1 1 1 -1 R = 1…

A: We will first find the matrix A using the QR factorization given.…

Q: Explain why S is not a basis for P2.S = {1 − x, 1 − x2, −1 − 2x + 3x2}

A:

Q: EX ample (12.1) Find the Singular Value De composiion of A= 0 4 3 4 00 3 00 Solution: FInd Q2…

A:

Q: Say whether the polynomials P1 (x) = x³–7x²+4x, p2(x) = 3x³-x²+5x-1, p3(x) = x³+4x-x²+1 are linearly…

A:

Q: Explain how fixing a basis v1, corresponding function in n variables of the form q(x1, · ·· , Xn) =…

A:

Q: 2. Find the basis for the nullspace of A, and the nullity and rank of A

A: This is a problem of linear transformation.

Q: 3. Let A {v1, v2, ..., Vn} be a basis for pV and n 2 2. Is B = {v1 + v2, , v2 + V3, ..., Vn + Vı}…

A:

Q: 6. Determine the basis for the null space of -2x + 2y + 2z] a) 3x + 5y + z 2y + z х — 2у — z b) -x +…

A: Introduction: Null Space, like Row and Column Space, is a simple area in a matrix, including all…

Q: [ 4 -1 Find a basis for the null space of A = - 20 {le Ex: 1

A: If x belongs to null space of a matrix A then Ax is equal to zero.

Q: If b1, b2 form an orthonormal basis for C and x is any vector in C, then O x4- x bị x b2 0 O x+ x bị…

A: We will find out the required value.

Q: (10) R¹3x+2y-20). Determine an orthonormal basis of W

A: Given : W={x,y,z∈R3:3x+2y-z=0}

Q: 4. A = 3 -6 -12 -3 -6 -9 -3 Find an orthonormal basis of the kernel of A.

A: Note: We know that the kernel of a matrix A is the set of solutions of the corresponding homogeneous…

Q: $5.5, Exercise 4. Find a basis for the null space of 0 -1 A = -1 4 -5 1 -2 What is the dimension of…

A: The given matrix is A=10-121-1004-51-2. First convert the above matrix into reduced row echelon form…

Q: Find a basis for Col B where [1 1 0 -2 0 0]

A:

Q: Extend {l + x, 1 + x + x2} to a basis for P 2.

A:

Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², –1 – 6x + 7x²} O s is linearly dependent. O…

A:

Q: which is the collection of all polynomials of degree ≤ 3. Write out the standard basis for P2? What…

A:

Q: Explain why S is not a basis for M, 2: -{::}[::} 0 1 1 S = 0 1 1 0 S is linearly dependent. s does…

A:

Q: Explain why S is not a basis for P,. .2 S = (1 - x, 1 - x², -1 – 8x + 9x²} O s is linearly…

A:

Q: Explain why S is not a basis for R2.S = {(1, 2), (1, 0), (0, 1)}

A:

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…

Q: a) Find the basis for the solution space of I1 +a2 - r3 = 0, -2r1 - 12 + 5r3 = 0, -11 + 4r3 = 0 %3D…

A: (a) given system of linear equation x1+x2-x3=0-2x1-x2+5x3=0-x1+4x3=0 to find - basis for solution…

Q: Explain why S is not a basis for P2.S = {−x, 4x2}

A: Given: S={−x, 4x2} To Explain: Why S is not a basis of P2.

Q: 9. Find a basis for the hyperplane a where a-(2, 1, -3).

A: Any two linearly independent vectors which are orthogonal to the given a are forms a orthogonal…

Q: 4 El 4 Let Vi V2 3 and V3 = Use the Gram-Schmidt procedure to construct an ordered - 4 orthonormal…

A: Concept introduction: Gram-Schmidt orthogonalization process Let V be a vector space with an inner…

Q: 8. Let W = Span{v1, v2}. Find an orthonormal basis for W. Vị = -2 , V1 = 4 2

A: An orthonormal basis for an inner product space V with finite dimension is a basis for V whose…

Q: 2 1. Find a basis of the null space of the matrix B -2 -2 -4

A:

Q: 1 1 11

A:

Q: 4 2. If a basis for R3 is B = -3 and given [x]B find i in the standard basis.

A:

Q: Suppose A E R4×3 has the following QR factorization A = = QR V2 1 -1 1 |1 -1 V2 -V2 [1 1 0 1 -1 0 0…

A: Here, we need to find an orthonormal basis of A. First we will row reduce A to find the column basis…

Q: Explain why S is not a basis for P,. S = {1 – x, 1 – x², -1 – 6x + 7x2} || O sis linearly dependent.…

A:

Q: Lemma 2.8 B is a basis of T iff

A: To prove that Β is a basis of Τ iff ∀Ο∈Τ ∃ Β0⊂Β: Ο=∪ ΒΟ

Q: 23. Suppose R* = Span {v1, ..., V4}. Explain why {v1,..., V4} is a basis for R4.

A:

Q: 2. Let S ={2 , 1 in M31. a. Determine whether S spans M31. b. Determine whether S is linearly…

A:

Q: 2. Construct a basis for T2,2(R²) and prove that it is a basis.

A: To find the basis of I2×2ℝ2. The set of all 2×2 matrices over ℝ. Let us claim that…

Q: find basis and the 1) dimension of w ={A ¢ Mnz: A +=0}

A:

Q: Let P2 be the polynomials of degree no more than 2. Is {x2+x+1,2x+1,3x2+1} a basis for P2 ?

A:

Q: - Let 3/2 U2 = U3 a. Show that {u1, u2, u3} is an orthonormal basis for R³.

A:

Q: a b 1) Let T cla + 2b + 1c + 3d + (-11) e+ (-la+ (-1) b + (-2) c + Od + 7e) x. d to Then a basis for…

A: The given transformation is  Tabcde=1a+2b+1c+3d+-11e+-1a+-1b+-2c+0d+7ex To find a basis for KerT⊥.

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.