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  1. Consider a set of five arbitrary 2x2 matrices. Can you always write one as a linear combination of the others? Explain. Repeat for five arbitrary 3x3 matrices.
  2. For each of the following sets explain whether or not the set is/could be a basis for the space mentioned. a) Four vectors in that form a loop when end-to-end connected.
    ...............

    Problem is i didnt understand the questions.
    For the first question:
    i know what linear combination is, but Do i have to find that five matrices? which can be linear combination of one another ? or is asking that whether any 5 matrices can be linear combination of one another or cant?
    For the second question:
    i have learned Standard basis of matrix, (actually learning) but there is no standard. Im just started learning and bit confused
    if any1 explain it little bit i will figure it out.
    Thanks
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2 Answers 2

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The first part of the first question is asking whether every set of five $2\times 2$ contains one that is a linear combination of the other four; the second part is asking the same question about sets of five $3\times 3$ matrices. The first part can be paraphrased as follows: is every set of five $2\times 2$ matrices linearly dependent? Similarly for the second part of the first question.

The second question is asking whether four geometric vectors that form a loop can be a basis. An example of such a loop in $\Bbb R^2$ is shown here:

enter image description here

HINT: What’s the sum of four vectors that form a loop?

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  • $\begingroup$ set of five means: 5 different matrices right ? not every elements are 5. <br> $\endgroup$
    – Andy
    Dec 17, 2013 at 6:40
  • $\begingroup$ @user3078437: Yes, a set of five matrices. They might not be different matrices, but there are five of them, and they can have any entries. $\endgroup$ Dec 17, 2013 at 6:41
  • $\begingroup$ as i understood First question`s answer: I cant because i can find five matrices which are cannot be linear combination of one another. for example ... and i will find example :) is it acceptable ? $\endgroup$
    – Andy
    Dec 17, 2013 at 7:00
  • $\begingroup$ @Andy: The two parts of the first question have different answers. In one case you can always write one of the five matrices as a linear combination of the others, so you need a proof of this, not an example. In the other case it is possible to find five matrices such that none of them is a linear combination of the others, and you need to find an example like that. HINT: What is the dimension of the space of $3\times 3$ matrices? $\endgroup$ Dec 17, 2013 at 7:08
  • $\begingroup$ then how about this answer ? : i can express 2x2 five matrix as one 4x5 matrix such that by the row reduction operation i can say(prove) that the 5th column(matrix) is linear combination of other matrices. $\endgroup$
    – Andy
    Dec 20, 2013 at 11:49
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The dimension of the vector space $V$ is the cardinality of a basis in this vector space and it's also the maximal number of linearly independant vectors of $V$ (or the minimal number of vectors which span $V$). Now what's the dimension of $\mathcal{M}_2(\Bbb R)$? of $\mathcal{M}_3(\Bbb R)$? Can you take it from here?

For your second question, what's the sum of Four vectors in that form a loop when end-to-end connected? What's this mean using the notion of linearly independant?

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  • $\begingroup$ Another well-deserved $\uparrow$! $\endgroup$
    – amWhy
    Dec 17, 2013 at 17:55

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