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Math 125 Exercise - 3 Linear Transformation: 1. Show that the following maping f are linear: f:R² → R² defined by f(r, y) = (r + y. 1). f : R³ → R defined by f(r, y, z) = 2r – 3y + 4z. f:R? +R defined by f(r, y) = ry.

Math 125 Exercise - 3 Linear Transformation: 1. Show that the following maping f are linear: f:R² → R² defined by f(r, y) = (r + y. 1). f : R³ → R defined by f(r, y, z) = 2r – 3y + 4z. f:R? +R defined by f(r, y) = ry.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Math 125 Exercise - 3 Linear Transformation: 1. Show that the following maping f are linear: f: R - R² defined by f(x, y) = (x + y, x). f: R* +R defined by f(r, y, z) = 2.r – 3y + 4z. • f : R? → R defined by f(r, y) = ry. • f : R² → R³ defined by f(x, y) = (r +1, 2y, r + y). f: R* - R² defined by f(x, y, z) = (|r|, 0). Linear Combination and Span of Vector Space and Sub Space: 2. Show that the ry plane W = (x, y,0) in R* is generated by: (i)u = and (ii)u = and v = 3. Show that that the vector u = -5 is a linear combination of the vectors v, = 3 2 -4 and v3 = -5 4. For which value of k the vector u = -2 is a linear combination of the vectors 3 and ty = of the vectors vi = and tz = Linearly In-dependency and Dependency 5. Determine whether or not the following vectors i R are linearly dependent: (1, -2, 1), (2, 1, –1), (7, -4, 1) (1, 2, –3), (1, –3, 2), (2, –1,5) (1, –3, 7), (2,0, –6), (3, –1, –1), (2,4, –5) (2, –3, 7), (0,0,0),, (3. -1, -4)
Transcribed Image Text:Math 125 Exercise - 3 Linear Transformation: 1. Show that the following maping f are linear: f: R - R² defined by f(x, y) = (x + y, x). f: R* +R defined by f(r, y, z) = 2.r – 3y + 4z. • f : R? → R defined by f(r, y) = ry. • f : R² → R³ defined by f(x, y) = (r +1, 2y, r + y). f: R* - R² defined by f(x, y, z) = (|r|, 0). Linear Combination and Span of Vector Space and Sub Space: 2. Show that the ry plane W = (x, y,0) in R* is generated by: (i)u = and (ii)u = and v = 3. Show that that the vector u = -5 is a linear combination of the vectors v, = 3 2 -4 and v3 = -5 4. For which value of k the vector u = -2 is a linear combination of the vectors 3 and ty = of the vectors vi = and tz = Linearly In-dependency and Dependency 5. Determine whether or not the following vectors i R are linearly dependent: (1, -2, 1), (2, 1, –1), (7, -4, 1) (1, 2, –3), (1, –3, 2), (2, –1,5) (1, –3, 7), (2,0, –6), (3, –1, –1), (2,4, –5) (2, –3, 7), (0,0,0),, (3. -1, -4)
Math 125 Exercise - 4 1. Determine whether or not the following vectors form a basis for the vector space R³: • (1, 1, 1) and (1, –1, 5). • (1, 2, 3), (1,0, -1), (3, –1,0) and (2, 1, –2). • (1, 1, 1), (1, 2, 3) and (2, –1, 1). • (1, 1, 2), (1,2, 5) and (5, 3, 4). 2. Let W be the subspace of R' generated by the vectors (1, –2, 5, –3), (2, 3, 1, –4) and (3, 8, –3, –5). find the basis and the dimension of W and the extend the basis of W to a basis of the whole space R. 3. Find the basis of columnspace and nullaspace of matrices: Г1 3 1 -2 -3] 2 -31 1 3 1 4 A = 3 -1 2 -2 -4 B = 1. 2 3 -4 -7 -3 and C = 3 -2 -1 5 -1 3 8 1 -7 -8 4 -2 and also find the rank and nullity of the matrices. 4. Find the Eigenvalues of the matrices: ГО -3 31 A = 3 -5 3 B = -3 1 -17 -7 5 -1 6. -6 4 -6 6 Also find the basis for eigen-space. Are the matrices diagonalizeable?
Transcribed Image Text:Math 125 Exercise - 4 1. Determine whether or not the following vectors form a basis for the vector space R³: • (1, 1, 1) and (1, –1, 5). • (1, 2, 3), (1,0, -1), (3, –1,0) and (2, 1, –2). • (1, 1, 1), (1, 2, 3) and (2, –1, 1). • (1, 1, 2), (1,2, 5) and (5, 3, 4). 2. Let W be the subspace of R' generated by the vectors (1, –2, 5, –3), (2, 3, 1, –4) and (3, 8, –3, –5). find the basis and the dimension of W and the extend the basis of W to a basis of the whole space R. 3. Find the basis of columnspace and nullaspace of matrices: Г1 3 1 -2 -3] 2 -31 1 3 1 4 A = 3 -1 2 -2 -4 B = 1. 2 3 -4 -7 -3 and C = 3 -2 -1 5 -1 3 8 1 -7 -8 4 -2 and also find the rank and nullity of the matrices. 4. Find the Eigenvalues of the matrices: ГО -3 31 A = 3 -5 3 B = -3 1 -17 -7 5 -1 6. -6 4 -6 6 Also find the basis for eigen-space. Are the matrices diagonalizeable?
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