Linear Algebra Basics: True/False Questions and Rank Properties

AI Thread Summary
A subspace of R^n with a basis of 5 vectors requires that n is greater than or equal to 5, confirming that the statement is true. This is because a basis of 5 vectors indicates a dimension of 5, which cannot exist in a space of lower dimension. In contrast, if a 6x7 matrix A has a rank of 3, it will have only 3 linearly independent column vectors, making the assertion that A^T has 5 independent columns false. Understanding the properties of rank is crucial for these determinations. Overall, clarity on the definitions and properties of subspaces and matrix rank is essential for accurate conclusions.
Tony11235
Messages
254
Reaction score
0
I have a few true/false/depends questions.

If a subspace of R^n has a basis consisting of 5 vectors then n is greater than equal to 5. I say it's true because 5 linearly independent vectors span R^5. Is that correct?

If the rank of a 6x7 matrix A is 3 then A^T has 5 linearly independent column vectors. I am not sure on this. Any help would be nice.
 
Physics news on Phys.org
The first one is false, because it is not true for n greater than 5.

Claude.
 
Tony11235 said:
I have a few true/false/depends questions.

If a subspace of R^n has a basis consisting of 5 vectors then n is greater than equal to 5. I say it's true because 5 linearly independent vectors span R^5. Is that correct?
Yes, it is true. If the subspace has a basis consisting of 5 vectors, then it has dimension 5. Certainly it can't be a subspace of Rn if n is less than 5 (but Rn is a subspace of itself). Notice that you must say "greater than or equal to 5" because we are talking about a subspace, not Rn itself.
(That's why Claude Bile's answer is incorrect. I suspect he confused the subspace with Rn itself.)

If the rank of a 6x7 matrix A is 3 then A^T has 5 linearly independent column vectors. I am not sure on this. Any help would be nice.
No, if the rank is 3 then it has 3 linearly independent column vectors. How did you get 5?
 
I didn't come up with 5, it was a true/false/depends question on my homework.
 
Okay, then it's false!
 
you need to go back and learn what rank means and what its properties are. otherwise even halls' (correct) statement will not do you much good.
 

Thread 'Struggling to understand the concept of this question (ice cube in water optics question)'

TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
View full post »
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
View full post »

Similar threads

I Confused about small detail in rank-nullity theorem
Replies
1
Views
2K
I Uniqueness of reduced row echelon form (proof verification)
Replies
4
Views
1K
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
Replies
3
Views
2K
I Dimension and solution to matrix-vector product
Replies
4
Views
3K
Help with linear algebra: vectorspace and subspace
Replies
15
Views
3K
I Characterizing linear independence in terms of span
Replies
3
Views
1K
I Spherically symmetric manifold
Replies
10
Views
841
I Row space, Column space, Null space & Left null space
Replies
8
Views
3K
Given specific v, dimension of subspace of L(V, W) where Tv=0?
Replies
15
Views
2K
I Vector subspace and basis vectors in the context of data science
Replies
6
Views
3K

Hot Threads

Recent Insights

Back
Top