Find all scalars c, if any exist, such that i+ cj + (c – 3)k is in the span of i+ 5j + 2k and - 5i + 25j + 10k. Show your work.

Q: Find the matrix A of the orthogonal projection onto the line L in R² that consists of all scalar…

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Q: Find the scalar product a•b when (i) a= x+2y -z and b=2x+3y +z (ii) a=x-3y +z and b=2x+y +z

A: We have to find the scalar product a·b when (i). a=x+2y-z and b=2x+3y+z(ii). a=x-3y+z and b=2x+y+z

Q: Consider the vector space P, of polynomials with real coefficients of degree at most two together…

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Q: Q1:Find an orthonormal basis for the Euclidean space Rª that contains the vectors -HH [0] u1 = ; Uz…

A: Here we have to find an orthonormal basis for the Euclidean space R4 that contains the vector…

Q: ) Find all scalars A such that A(2,3) has length 1.

A: Given query is to find the value of scalars for which length of given vector is 1.

Q: (a) Prove that if u is orthogonal to both v and w, then u is orthogonal to v + w. (b) Prove that if…

A: Given: (a) If u is orthogonal to both v and w. To prove:  u is orthogonal to v + w. (b) If u is…

Q: Let f E P(R) be a polynomial of degree n. Let yo,..., Yn be the dual basis of 1, x, x²,..., x". Show…

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Q: 3. (a) Show that f(r) = 1- are orthogonal elements of C[0, 1] with respect to the = z and g(a) inner…

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Q: The set of all polynomials of degree 6 under the standard addition and scalar multiplication…

A: Polynomials of degree n is a set which is not closed under addition. For example, if n=3, then  x3…

Q: is not diagonalizable (by similarity) but is condiagonalizable. 4.6.P2 Show that (a) [ (b) [] is…

A: To show: (a) The matix i10i is not diagonalizable but it is condiagonalizable. (b) The matix 1-111…

Q: Prove that if A is invertible and diagonalizable, then A-1 is also diagonalizable.

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Q: 5. Show that if S = {v1,V2, .. .} is a basic for the space vector V and c is a non-zero ... ... ...…

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Q: The set of all polynomials of degree 6 under the standard addition and scalar multiplication…

A: The answer is given below

Q: 8. Let x and y be vectors in 3-space, and suppose u is orthogonal to both x and y. Prove that u is…

A: Let x and y be vectors in 3-space and suppose u is orthogonal to both x and y  This implies  < x,…

Q: . u. ): Is the set S = {r +x-1,22 - r+3,-a+3z-2}) a base for the Pa(r) vector space which is a…

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Q: 1: Find a basis and determine the dimension of the following sub-spaces. H = { : 1a-c=0}. a b {:…

A: We have to find the basis and dimension of subspace

Q: In the vector space P, of polynomials of degree at most two, the coordinates of the vector p(x)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Is the set of polynomials {-1+ 3t – 21², 4 – 10t + 312, 4 – 8t – 212} independent? Answer this…

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Q: Q1:Find an orthonormal basis for the Euclidean space R* that contains the vectors U1 uz = and uz

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Q: Find an algebraic description of the range of A by row reducing A^T and using the relation between…

A: We know that rank(A) = rank(AT). We need to find an algebraic description of the range of A by row…

Q: (b) Find the orthogonal decomposition of v with respect to W. projw(v): %3D perpw(v)

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Q: (a) Find scalars a and b such that z = ax + by. (a, b) = (b) Do there exist scalars a and b such…

A: Please find the solution

Q: This is a textbook question, not a graded question   The first four Hermite polynomials of the…

A: We  have to find the basis for P3

Q: (a) Find the coordinate vector of æ with respect to the ordered basis E = -2 of R3:

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Q: Consider P2(R) with its basis B = (x, x + 1, x² –- 1). Find [x2 + 3x + 5]B.

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Q: Find the matrix A representing the reflection about the line L in R² which consists of all scalar…

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Q: The inner product, (a|b), of any two vectors a and b in a vector space must satisfy the conditions…

A: Here inner product given by  (aIb) =a1×b1+12a1×b2+a2×b1+a2×b2   where a= (a1,a2 )  and b = (b1,b2)…

Q: Expand x = (a, b, c, d) with respect to the orthogonal basis of: 2 -1 1 Vị = V2 -2 3 Express your…

A: Note: Observe that the given orthogonal basis spans only two dimension subspace of ℝ4, therefore the…

Q: Let u = <1, 2, 1> , v = <1, -1, -1> , w = <1, 1, -1> , and z = <2, -3, -4> .…

A: Given vectors

Q: If o is a scalar point function and c is a scalar then prove that grad (c o) = c (grad o). %3D

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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: The set of all polynomials of degree † under the standard addition and scalar multiplication…

A: Suppose  P7=f(x) | f(x) is a polynomial of degree 7. Then we have to check why this set is not a…

Q: Prove that the scalar 1 is the identity for scalar multiplication: 1A = A.

A: To prove that 1 is the identity element for scalar multiplication of matrix A. Let there is a matrix…

Q: In the vector space of polynomials of degree at most two, coordinates of the vector p()=5+x+5x2 in…

A: Given p=5+x+5x2and Basis B={e1 =1+x, e2=x+x2 ,e3=2+x}

Q: Recall that the Euclidean scalar product in R² is defined by (x1, Y1) · (x2, Y2) := X1T2+ Y1Y2 - (i)…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: Use the inner product (f, 9) = f(2)g(=) dæ in the vector space P(R) of polynomials to find (f, 9),…

A: The inner product is f, g=∫01 f(x)g(x) dx The functions given are f(x)=10x2-3 and g(x)=-9x+10

Q: Is the set of vectors v = 6, V2 =5i form a basis for the set of all complex numbers of the form a +…

A: To check the basis of the vectors of the set of complex numbers of the form a+ib. Check for the…

Q: Find the orthogonal decomposition of v with respect to W

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Q: Consider the basis B= {1, x+x2, 2x2-1} for the vector space P,. The coordinate vector of the…

A: Given that the basis B={1,x+x2,2x2-1} for the vector space P2. Here we have to find the coordinate…

Q: /- X be the vector Space of all Complex 2x2 and de fine T: x →x by Tx = bx, where Let matrices bEX…

A: Given :-  Let X be the vector space of all complex 2×2 matrices and define T : X →X  by Tx = bx ,…

Q: Consider the vector space P, with the standard inner product. Calculate || p || and d(p,q) for the…

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Q: Let V = span {V1, V2}, where V1 and V2 = 1 2 Find a basis S for R3 that includes vi and v2.

A: Let V  = span { v1 , v2}  where   v1= 102 and v2 = 111 Find a basis S for R3 that includes v1 and…

Q: Find a basis for the plane 4x + 3y + 5z = 0 in R³.

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Q: Let B = be any 2 x 2 matrix. COS a (i) Show that there are real numbers un and a such that u11 sin a…

A: Given: A 2×2 matrix, B=abcd

Q: Find the norm of the image of U(4.9, 5.1) by a rotation of 0 = A/1o Answer:

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Q: In the vector space P, of polynomials of degree at most two, the coordinates of the vector…

A: The basis of the given polynomial is: B= e1=1+2x,  e2=x-3x2,  e3=1+x2

Q: 9. Let i, j denote the standard basis in R2. Find a scalar c if the vector 7i- 2j is perpendicular…

A: Dot product of the two vectors.

Q: Consider the basis {1, 1 - t, 1 + t + t2} of the space P2 of polynomials of degree at most two. Find…

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