Let Tφ: R2 → R2 be the transformation which rotates the plane by an angle φ Determine the angle between the green arrows and prove that Tφ is a linear transformation.
1) Let Tφ: R2 → R2 be the transformation which rotates the plane by an angle φ Determine the angle between the green arrows and prove that Tφ is a linear transformation.
2) Let T be a linear transformation. Prove that if {T(¯v1), . . . , T(¯vn)} are linearly independent then {v¯1, . . . , v¯n} are linearly independent.
3) Find the standard matrix for the linear transformation Tθ, φ: R 3 → R 3 which rotates a vector by the angle θ in the x1x2-plane and by the angle φ in the x1x3-plane
4) Let A be a m × n matrix and B a n × p matrix. Prove that (AB) T = BT AT
5) Prove that the statement AB = O implies B = O is equivalent to A being invertible. (Hint: look at the columns)
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