1. Introduction To Linear Algebra Pdf Download
2. Introduction To Linear Algebra 5th Edition Pdf Download Mac

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Introduction To Linear Algebra Pdf Download

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Introduction To Linear Algebra 5th Edition Pdf Download Mac

• Step 1 of 9

  We need to describe geometrically (line, plane or allof)all linear combinations of the given vectors.

  We know that the linear combination of two vectors v and w isgiven by,where c and d are two scalars.

• Step 2 of 9

  (a) Let us considerthe following vectors

  and

  First of all we can notice that the vector w can be obtained by multiplying the vectorv by the scalar 3.

  Thus, we have

  This means that both the two vectors lie along the sameline.

• Step 3 of 9

  Next, we know that the linear combination of two vectors can beexpressed as,where c and d are two scalars.

  Now, since the two vectors lie along the same line, it can beconsidered that their linear combinations are effectively a set oflinear combinations of a single vector.

• Step 4 of 9

  This will give us an infinitely long line of vectors(say)which may be forward or backward.

  Thus, all the linear combinations of the vectors and occupy a line in.

• Step 5 of 9

  (b) Let us consider the following vectors

  and

  We notice that the two vectors v and w donot lie along the same line.

  Also, their linear combination can be expressed as,where c and d are two scalars.

• Step 6 of 9

  Thus, in this case we will be adding two vectors that are alongdifferent lines.

  We can easily notice that adding on one line to all on the other line will fill the two-dimensional region between thetwo vectors.

  Thus, all the linear combinations of the vectors and occupy a plane in.

• Step 7 of 9

  (c) Let us consider the following vectors

  ,

  We notice that no pair of vectors lie along the same line.

  Also, their linear combination can be expressed as,where c, d and e are scalars.

• Step 8 of 9

  An easier way to proceed is to first combine two vectors and addthe third vector.

  Let us first consider the linear combinations of vectorsu and v.

  We can easily notice that adding on one line to all on the other line will fill the two-dimensional region between thetwo vectors.

  This gives us a plane in.

• Step 9 of 9

  Finally, we combine all to the plane obtained.

  This gives a three-dimensional space covered by vectors (forwardand backward).

  Thus, all the linear combinations of the vectors, and occupy all of.