Question

# Let A denote an$m \times n$matrix. (a) Show that null A=null(UA) for every invertible$m \times n$matrix U. (b) Show that dim(null A)=dim(null(AV)) for every invertible$n \times n$matrix V. [Hint: If$\left\{ \mathbf { x } _ { 1 } , \mathbf { X } _ { 2 } , \dots , \mathbf { X } _ { k } \right\}$is a basis of null A, show that$\left\{ V ^ { - 1 } \mathbf { x } _ { 1 } , V ^ { - 1 } \mathbf { x } _ { 2 } , \dots , V ^ { - 1 } \mathbf { x } _ { k } \right\}$is a basis of null(AV).]

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