1. (a) Solve the system: x1 + x2 − x3 − 2x 4 + x5 = 1 2x1 + x2 + x3 +
... 16. Suppose that T : V1 → V2 is a one-to-one linear transformation and suppose that H is a nonzero subspace of the vector space V1 . Then T (H), the set of all images of vectors in H under T , is a subspace of V2 . (a) Define what it means for a set B = {v1 , v2 , ..., vn } to be a basis for H. (b) ...
... 16. Suppose that T : V1 → V2 is a one-to-one linear transformation and suppose that H is a nonzero subspace of the vector space V1 . Then T (H), the set of all images of vectors in H under T , is a subspace of V2 . (a) Define what it means for a set B = {v1 , v2 , ..., vn } to be a basis for H. (b) ...
Solution - UC Davis Mathematics
... Solution: (a) No: it does not contain the identity! (b) Yes, it is a normal subgroup. It contains the identity, the product of two rotations is a rotation and the inverse of a rotation is a rotation, so it is a subgroup. Also, det(g 1 xg) = det(x), so any matrix conjugate to a rotation must have det ...
... Solution: (a) No: it does not contain the identity! (b) Yes, it is a normal subgroup. It contains the identity, the product of two rotations is a rotation and the inverse of a rotation is a rotation, so it is a subgroup. Also, det(g 1 xg) = det(x), so any matrix conjugate to a rotation must have det ...
2.5 Multiplication of Matrices Outline Multiplication of
... However, multiplication of matrices is not done entrywise. It turns out that when we are dealing with data matrices, entrywise multiplication often does not give us the information we are looking for. Given two matrices A and B, we write AB for the matrix A × B. To multiply two matrices, the number ...
... However, multiplication of matrices is not done entrywise. It turns out that when we are dealing with data matrices, entrywise multiplication often does not give us the information we are looking for. Given two matrices A and B, we write AB for the matrix A × B. To multiply two matrices, the number ...
More Selected Solutions 7A-B Multiplication by a complex number is
... Multiplication by w is an isometry if the distance between wz1 and wz2 is equal to the distance between z1 and z2 for any complex numbers z1 and z2 . |wz1 − wz2 | = |w(z1 − z2 )| = |w| |z1 − z2 |, so this equals |z1 − z2 | if and only if |w| = 1, or in other words a2 + b2 = 1. Note this is also the ...
... Multiplication by w is an isometry if the distance between wz1 and wz2 is equal to the distance between z1 and z2 for any complex numbers z1 and z2 . |wz1 − wz2 | = |w(z1 − z2 )| = |w| |z1 − z2 |, so this equals |z1 − z2 | if and only if |w| = 1, or in other words a2 + b2 = 1. Note this is also the ...
