Homework Answers
Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2...
Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2 + x4 = 0 and x2 + x3 + x4 = 0 X3 X4 = Find an orthonormal basis H {h1, h2, h3, h4} for R4 with the property that hy and h2 are elements of an orthonormal basis for W, where orthonormality is defined with respect to the dot product on R4 x R4.
Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2...
Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2 + x4 = 0 and x2 + x3 + x4 = 0 X3 X4 = Find an orthonormal basis H {h1, h2, h3, h4} for R4 with the property that hy and h2 are elements of an orthonormal basis for W, where orthonormality is defined with respect to the dot product on R4 x R4.
The set x1, x2} is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthonormal basis for W exactly...
The set x1, x2} is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthonormal basis for W exactly as described in the book. Instructions: You must perform the process by using the first vector in the list as X1 and the second vector as x2. The answer is unique! Round your answer to three decimal places. 3 2 1 -1 -9 X1 X2= -6 -6 0.309 0.154 V1 V2 -0.154 -0.926
The set x1, x2}...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4 2 5 and x2 ܕ ܩ ܟ 6 -7 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4 2 5 and x2 ܕ ܩ ܟ 6 -7 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and...
0 5 The set of vectors {x1, x2} spans a subspace W of R3, where x1...
0 5 The set of vectors {x1, x2} spans a subspace W of R3, where x1 = 19- and X 2 -- 2 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and normalize it...
Exercise 1. The set {x1,x2} is a basis for a subspace W. Use the Gram-Schmidt process...
Exercise 1. The set {x1,x2} is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. Hint: Scaling vectors before you begin may simplify calculations.
1. Prove that each of the following is a subspace. (a) W = {x: x =...
1. Prove that each of the following is a subspace. (a) W = {x: x = (x 1, 22, 23) and X1 + 12 = x;} (b) W = {p: p(t) = ata + b + c and a+b+c=0} (C) W = {A € R2x2 and A is upper triangular) (d) W = {f:f EC(0,1) and f(0 =0} 2. Show that the following subsets of A R2x2 are not subspaces. (a) W = {A : A is the singular matrix}...
Let V⊂R^4 be the subspace defined by the equation x1 + 3x2 - 5x3 - x4 = 0. a) Find an orthogonal basis for V. b) Which is the point over the plane x1 + 3x2 - 5x3 - x4 = 36 closest to the origin?
Let V⊂R^4 be the subspace defined by the equation x1 + 3x2 - 5x3 - x4 = 0. a) Find an orthogonal basis for V. b) Which is the point over the plane x1 + 3x2 - 5x3 - x4 = 36 closest to the origin?
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0},...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2 + x4 = 0 and x2 + x3 + x4 = 0 X3 X4 = Find an orthonormal basis H {h1, h2, h3, h4} for R4 with the property that hy and h2 are elements of an orthonormal basis for W, where orthonormality is defined with respect to the dot product on R4 x R4.
Consider the subspace W C R4 given by X1 X2 W = ER4 X1 + x2 + x4 = 0 and x2 + x3 + x4 = 0 X3 X4 = Find an orthonormal basis H {h1, h2, h3, h4} for R4 with the property that hy and h2 are elements of an orthonormal basis for W, where orthonormality is defined with respect to the dot product on R4 x R4.
The set x1, x2} is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthonormal basis for W exactly as described in the book. Instructions: You must perform the process by using the first vector in the list as X1 and the second vector as x2. The answer is unique! Round your answer to three decimal places. 3 2 1 -1 -9 X1 X2= -6 -6 0.309 0.154 V1 V2 -0.154 -0.926
The set x1, x2}...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4 2 5 and x2 ܕ ܩ ܟ 6 -7 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4 2 5 and x2 ܕ ܩ ܟ 6 -7 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and...
0 5 The set of vectors {x1, x2} spans a subspace W of R3, where x1 = 19- and X 2 -- 2 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and normalize it...
Exercise 1. The set {x1,x2} is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. Hint: Scaling vectors before you begin may simplify calculations.
1. Prove that each of the following is a subspace. (a) W = {x: x = (x 1, 22, 23) and X1 + 12 = x;} (b) W = {p: p(t) = ata + b + c and a+b+c=0} (C) W = {A € R2x2 and A is upper triangular) (d) W = {f:f EC(0,1) and f(0 =0} 2. Show that the following subsets of A R2x2 are not subspaces. (a) W = {A : A is the singular matrix}...
1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of...
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