Exercises 31 and 32 reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T : V → W be a linear transformation, and let { v 1 ,..., v p } be a subset of V . 32. Suppose that T is a one-to-one transformation, so that an equation T ( u ) = T ( v ) always implies u = v . Show that if the set of images { T ( v 1 ),…, T ( v p )} is linearly dependent, then { v 1 ,..., v p } is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set (because in this case the set of images cannot be linearly dependent).
Exercises 31 and 32 reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T : V → W be a linear transformation, and let { v 1 ,..., v p } be a subset of V . 32. Suppose that T is a one-to-one transformation, so that an equation T ( u ) = T ( v ) always implies u = v . Show that if the set of images { T ( v 1 ),…, T ( v p )} is linearly dependent, then { v 1 ,..., v p } is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set (because in this case the set of images cannot be linearly dependent).
Solution Summary: The author explains that linear transformations are distributive and associative. If set of images is linearly dependent in W, then leftt (v_1),do
Exercises 31 and 32 reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let T : V → W be a linear transformation, and let {v1,...,vp} be a subset of V.
32. Suppose that T is a one-to-one transformation, so that an equation T(u) = T(v) always implies u = v. Show that if the set of images {T(v1),…,T(vp)} is linearly dependent, then {v1,..., vp} is linearly dependent. This fact shows that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set (because in this case the set of images cannot be linearly dependent).
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
Algebra 1, Homework Practice Workbook (MERRILL ALGEBRA 1)
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Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY