Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation....

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation. We say a linear transformation ?:?→? is a left inverse of ? if ST=I_v, where ?_v denotes the identity transformation on ?. We say a linear transformation ?:?→? is a right inverse of ? if ??=?_w, where ?_w denotes the identity transformation on ?. Finally, we say a linear transformation ?:?→? is an inverse of ? if it is both a left and right inverse of ? . When ? has an inverse, we say ? is invertible.

Show that

(a) ? has a left inverse iff ? is injective.

(b) If ? is a basis for ?V and ? is a basis for ?, then [?]^?_?(Transformation from basis ? to ?) has a left inverse iff its columns are linearly independent.

Solutions

Expert Solution

Colby Messinger answered 1 year ago

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