Question

3.[4p] (a) In the following questions assume that a linear operator acts from a finite- dimensional linear space X to X, and assume that the word "vector means an element of X. Recall that a vector a is a pre-image of a vector y (and y is the image of x) for a linear operator A: X -> X, if Ax-y. How many of the following statements are true? (i) A linear operator maps a basis into a basis. (ii) The set of all vectors, which are mapped to the zero vector, is a linear subspace. ear operator maps the zero vector to the zero vector (iv) A linear operator maps a complete set of vectors into a complete set of vectors. (v) A linear operator maps a linearly independent set of vectors into a linearly indepen dent set of vectors. (b) In the following questions assume that a linear operator acts from a finite- dimensional linear space X to X, and assume that the word "vector" means an element of X. Recall that a vector z is a pre-image of a vector y (and y is the image of x) for a linear operator A: X → X, if Ax-y. How many of the following statements are true? (i) For any linear operator every vector has a pre-image (ii) A lincar operator maps the zero vector to the zero vector. (ii) The set of all vectors, which have a pre-image vector, is a near subspace. (iv) For any linear operator every vector is co-linear with its image. (v) The set of all vectors, which are mapped to the zero vector, is a linear subspace

Answer 1

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