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1 Section A homogeneous equation is always consistent. TRUE - The trivial solution is always a solution. The equation Ax = 0 gives an explicit descriptions of its solution set. FALSE - The equation gives an implicit description of the solution set. The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. FALSE - The trivial solution is always a solution to the equation Ax = 0. The equation x = p + tv describes a line through v parallel to p. False. The line goes through p and is parallel to v. The solution set of Ax = b is the set of all vectors of the form w = p + v h where v h is any solution of the equation Ax = 0 FALSE This is only true when there exists some vector p such that Ap = b.
2 Section If x is a nontrivial solution of Ax = 0, then every entry in x is nonzero. FALSE. At least one entry in x is nonzero. The equation x = x 2 u + x 3 v, with x 2 and x 3 free (and neither u or v a multiple of the other), describes a plane through the origin. TRUE The equation Ax = b is homogeneous if the zero vector is a solution. TRUE. If the zero vector is a solution then b = Ax = A0 = 0. So the equation is Ax = 0, thus homogenous. The effect of adding p to a vector is to move the vector in the direction parallel to p. TRUE. We can also think of adding p as sliding the vector along p. The solution set of Ax = b is obtained by translating the solution set of Ax = 0. FALSE. This only applies to a consistent system.
3 Section The columns of the matrix A are linearly independent if the equation Ax = 0 has the trivial solution. FALSE. The trivial solution is always a solution. If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S. FALSE- For example, [1, 1], [2, 2] and [5, 4] are linearly dependent but the last is not a linear combination of the first two. The columns of any 4 5 matrix are linearly dependent. TRUE. There are five columns each with four entries, thus by Thm 8 they are linearly dependent. If x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span{x, y}. TRUE Since x and y are linearly independent, and {x, y, z} is linearly dependent, it must be that z can be written as a linear combination of the other two, thus in in their span.
![FALSE- For example, [1, 1], [2, 2] and [5, 4] are linearly dependent but the last is not a linear combination of the first two. The columns of any 4 5 matrix are linearly dependent. TRUE.](/docs-images/46/21161837/images/page_3.jpg "There are five columns each with four entries, thus by Thm 8 they are linearly dependent. If x and y are linearly independent, and if {x, y, z} is linearly dependent, then z is in Span{x, y}.")
4 Section Two vectors are linearly dependent if and only if they lie on a line through the origin. TRUE. If they lie on a line through the origin then the origin, the zero vector, is in their span thus they are linearly dependent. If a set contains fewer vectors then there are entries in the vectors, then the set is linearly independent. FALSE For example, [1, 2, 3] and [2, 4, 6] are linearly dependent. If x and y are linearly independent, and if z is in the Span{x, y} then {x, y, z} is linearly dependent. TRUE If z is in the Span{x, y} then z is a linear combination of the other two, which can be rearranged to show linear dependence.
5 Section Continued If a set in R n is linearly dependent, then the set contains more vectors than there are entries in each vector. False. For example, in R 3 [1, 2, 3] and [3, 6, 9] are linearly dependent.
6 Section A linear transformation is a special type of function. TRUE The properties are (i) T (u + v) = T (u) + T (v) and (ii) T (cu) = ct (u). If A is a 3 5 matrix and T is a transformation defined by T (x) = Ax, then the domain of T is R 3. FALSE The domain is R 5. If A is an m n matrix, then the range of the transformation x Ax is R m FALSE R m is the codomain, the range is where we actually land. Every linear transformation is a matrix transformation. FALSE. The converse (every matrix transformation is a linear transformation) is true, however. We (probably) will see examples of when the original statement is false later.
7 Section Continued A transformation T is linear if and only if T (c 1 v 1 + c 2 v 2 ) = c 1 T (v1) + c 2 T (v 2 ) for all v 1 and v 2 in the domain of T and for all scalars c 1 and c 2. TRUE If we take the definition of linear transformation we can derive these and if these are true then they are true for c 1, c 2 = 1 so the first part of the definition is true, and if v = 0, then the second part if true.
8 Section Every matrix transformation is a linear transformation. TRUE To actually show this, we would have to show all matrix transformations satisfy the two criterion of linear transformations. The codomain of the transformation x Ax is the set of all linear combinations of the columns of A. FALSE The If A is m n codomain is R m. The original statement in describing the range. If T : R n R m is a linear transformation and if c is in R m, then a uniqueness question is Is c is the range of T. FALSE This is an existence question. A linear transformation preserves the operations of vector addition and scalar multiplication. TRUE This is part of the definition of a linear transformation. The superposition principle is a physical description of a linear transformation. TRUE The book says so. (page 77)
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