Section 1.5 23

I A homogeneous equation is always consistent. TRUE - Thetrivial solution is always a solution.

I The equation Ax = 0 gives an explicit descriptions of itssolution set. FALSE - The equation gives an implicitdescription of the solution set.

I The homogeneous equation Ax = 0 has the trivial solution ifand only if the equation has at least one free variable. FALSE- The trivial solution is always a solution to the equationAx = 0.

I The equation x = p + tv describes a line through v parallel top. False. The line goes through p and is parallel to v.

I The solution set of Ax = b is the set of all vectors of the formw = p + vh where vh is any solution of the equation Ax = 0FALSE This is only true when there exists some vector p suchthat Ap = b.

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Section 1.5 24

I If x is a nontrivial solution of Ax = 0, then every entry in x isnonzero. FALSE. At least one entry in x is nonzero.

I The equation x = x2u + x3v, with x2 and x3 free (and neitheru or v a multiple of the other), describes a plane through theorigin. TRUE

I The equation Ax = b is homogeneous if the zero vector is asolution. TRUE. If the zero vector is a solution thenb = Ax = A0 = 0. So the equation is Ax = 0, thushomogenous.

I The effect of adding p to a vector is to move the vector in thedirection parallel to p. TRUE. We can also think of adding pas sliding the vector along p.

I The solution set of Ax = b is obtained by translating thesolution set of Ax = 0. FALSE. This only applies to aconsistent system.

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Section 1.7 21

I The columns of the matrix A are linearly independent if theequation Ax = 0 has the trivial solution. FALSE. The trivialsolution is always a solution.

I If S is a linearly dependent set, then each vector is a linearcombination of the other vectors in S . FALSE- For example,[1, 1] , [2, 2] and [5, 4] are linearly dependent but the last isnot a linear combination of the first two.

I The columns of any 4× 5 matrix are linearly dependent.TRUE. There are five columns each with four entries, thus byThm 8 they are linearly dependent.

I If x and y are linearly independent, and if {x, y, z} is linearlydependent, then z is in Span{x, y}. TRUE Since x and y arelinearly independent, and {x, y, z} is linearly dependent, itmust be that z can be written as a linear combination of theother two, thus in in their span.

Linear Algebra, David Lay Week Three True or False

Section 1.7 22

I Two vectors are linearly dependent if and only if they lie on aline through the origin. TRUE. If they lie on a line throughthe origin then the origin, the zero vector, is in their span thusthey are linearly dependent.

I If a set contains fewer vectors then there are entries in thevectors, then the set is linearly independent. FALSE Forexample, [1, 2, 3] and [2, 4, 6] are linearly dependent.

I If x and y are linearly independent, and if z is in theSpan{x, y} then {x, y, z} is linearly dependent. TRUE If z isin the Span{x, y} then z is a linear combination of the othertwo, which can be rearranged to show linear dependence.

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Section 1.7 22 Continued

I If a set in Rn is linearly dependent, then the set contains morevectors than there are entries in each vector. False. Forexample, in R3 [1, 2, 3] and [3, 6, 9] are linearly dependent.

Linear Algebra, David Lay Week Three True or False

Section 1.8 21

I A linear transformation is a special type of function. TRUEThe properties are (i) T (u + v) = T (u) + T (v) and (ii)T (cu) = cT (u).

I If A is a 3× 5 matrix and T is a transformation defined byT (x) = Ax, then the domain of T is R3. FALSE The domainis R5.

I If A is an m × n matrix, then the range of the transformationx 7→ Ax is Rm FALSE Rm is the codomain, the range is wherewe actually land.

I Every linear transformation is a matrix transformation.FALSE. The converse (every matrix transformation is a lineartransformation) is true, however. We (probably) will seeexamples of when the original statement is false later.

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Section 1.8 21 Continued

I A transformation T is linear if and only ifT (c1v1 + c2v2) = c1T (v1) + c2T (v2) for all v1 and v2 in thedomain of T and for all scalars c1 and c2. TRUE If we takethe definition of linear transformation we can derive these andif these are true then they are true for c1, c2 = 1 so the firstpart of the definition is true, and if v = 0, then the secondpart if true.

Linear Algebra, David Lay Week Three True or False

Section 1.8 22

I Every matrix transformation is a linear transformation. TRUETo actually show this, we would have to show all matrixtransformations satisfy the two criterion of lineartransformations.

I The codomain of the transformation x 7→ Ax is the set of alllinear combinations of the columns of A. FALSE The If A ism × n codomain is Rm. The original statement in describingthe range.

I If T : Rn → Rm is a linear transformation and if c is in Rm,then a uniqueness question is ”Is c is the range of T .” FALSEThis is an existence question.

I A linear transformation preserves the operations of vectoraddition and scalar multiplication. TRUE This is part of thedefinition of a linear transformation.

I The superposition principle is a physical description of a lineartransformation. TRUE The book says so. (page 77)

Linear Algebra, David Lay Week Three True or False